Did the Jewish Temple influence Nazi architecture?

No.

Apart from Solomon's Temple being just unavailable as an actual model of any ascertainable concreteness (and Solomon being in all probability a fictional character to begin with [Finkelstein/Silberman: David and Solomon]), this 'cultural appropriation' was quite out of proportion for anything the Nazis wanted to adopt. For some very few examples a very indirect comparison might be made; but relating to an even earlier style from roughly that region. And with quite some stretching: 'that' 'ethnic heritage' — although that is then clearly adopting Nazi thinking in grouping people together and attributing such characteristics over such long periods of time. Re-inforcing again the initial: No.

The picture in question as the supposed 'model' is clearly the Herodian Jerusalem Temple. This has the added problem, that it was much less 'Jewish' in style, but clearly for the most part Hellenistic (column arcades, basilica at one side. Confer David. M. Jacobson: "Herod’s ‘Roman’ Temple", BAR 28.2, 2002, 18-27, 60-61.).
Modern reconstruction attempts and 'replicas' show the difficulties in arriving at such conclusions over details above ground:

In an innteresting temporal twist the Temple reconstruction in question is from an Israeli archaeologist, developed in 1966. Earlier and later attempts to depict reconstructions look like this:

And even more strikingly n contrast, the depiction in a mosaic in Dura-Europos:

So while the observations about peculiarities within the question are not to be dismissed outright in all specifics, the concrete example reverses the timeline. Rarely an attempt to depict the original second (Herodian) Temple looks that much like the picture presented.

It is further untrue that 'Nazi architecture' really is one unified style, but an umbrella term for a convoluted number of styles that are supposed to define an era. When in reality a number of architects and many styles were seen and only in the post-mortem of the regime (many architects just continued, some even in their very own style from before) lumped together into a temporal category, instead of really an 'artistic' or stylistic one.

The most rigid guidelines on where to look for inspiration are of course found in the SS, which sported not so few architects:

Such optimism for the future expansion of the SS and Germany in general translated as well into the broad architectural claims made in the official periodical for the SS, Das Schwarze Korps (1937–45). In this periodical, discussions of architecture were used to give SS administrators and architects design ideas from projects approved by the central office in Berlin and also to solidify an institutional identity through the promotion of particular architectural styles and examples. Articles were neither consistent nor accurate in their reference to monuments and epochs but were meant to inspire the SS membership and be used as a model for institutional architects. Most often authors cited classical architecture (particularly that of the Roman Empire), but they also discussed ancient Greece, Egypt and medieval German architecture as paradigms of previous architectural traditions. While relying on semi-accurate historical accounts or superficial formal comparisons, the buildings illustrated were invariably explained with the thesis that powerful dynasties will build the best architecture. This thesis would be repeated in multiple articles. SS authors described architecture not merely as the result of political, military or “racial” dominance, but as expressing and actively perpetuating power. When speaking of the eighteenth and nineteenth centuries, for example, an anonymous author claimed:

Exactly as the “Prussian Style” fundamentally signified nothing less than a revelation of Nordic thought, so the master sensibility (which, at that time, was especially strengthened politically) of the Germanic ruling class in England and Scandinavia used as well classical forms. This sensibility never acquired, significantly speaking, the usual slavish copying of the antique typical particularly in France, but rather stubbornly persevered to the spatial sentiment specific to our race.13

Yet the interest of the SS in using ideological pronouncements on architecture to promote a continuity between previous powerful epochs and its own present strength went beyond pronouncements in its magazine and the construction of the various institutional structures under its control.

_{— Paul B. Jaskot: "The Architecture of Oppression: The SS, Forced Labor and the Nazi Monumental Building Economy", Architext, Routledge, 2000.}

Such "powerful epochs" are completely incompatible with taking anything 'Jewish' as a model for anything else.

Or in other words:

We might ask, then, why the monuments erected under National Socialism to "incarnate"the community of the German people were imitations of the styles of the past, unlike the vernacular style adopted for youth hostels and model farms or the functionalism adopted for factories. Even though neoclassical style took over official architecture in both Europe and the United States during the 1930s, it was something more than fashion that explains why the German community was incarnated in imitations of Athens or of Babylon, of Frederick II's Castel del Monte or of the Roman Colosseum. It is not only to allow the community to identify with the "eternal soul of the race" by appropriating scattered fragments of it. In this instance, to imitate the styles of the past is to make something into the past, to pass something off and to make it pass away, to give visible and tangible proof, in the here and now, of the grandeur of a community that has become a work of art. It is to give proof of the power of a community incarnated as past, at the same time already there and already historical.

_{— Eric Michaud: "National Socialist Architecture as an Acceleration of Time", Critical Inquiry, 19, 1993.}

If there are common elements in monumental Nazi architecture to observe, then in stark contrast to seldom modernist, but an often also rustic, very much local, supposedly 'Germanic', style, then it would be a play on neo-classicism:

German scholar Rolf Badenhausen argues, “To call [Nazi architecture] classicist is not enough. Something new has been created.” The conglomeration of cultures and styles shares “a specific handwriting which makes them instantly recognizable as the product of the Third Reich.” Some of these unifying elements include stripped-down porticoes, stark rectilinear lines with a focus on heavy horizontals, rows of windows inset in heavy walls, an obsession with symmetry and repetition, and above all the element of simplicity. Hitler often remarked, “To be German means to be clear.”

Still, it cannot be denied that, along with his use of these modern elements, Hitler did have a penchant for imitating ancient edifices such as those found in Italy. This replication stemmed from his admiration of classical architecture. The first time Hitler traveled to Rome, he could only exclaim, “Rome bowled me over!”

Hitler could hardly admire a city as deeply as he did Rome and not want to emulate it or even surpass it in beauty and mass. Thus, he began to strive for a Roman Germany, declaring in his first speech as Chancellor, “…each politically historical epoch searches in its art for the link with a period of equally heroic past.” He found his in Rome.

Consequently, the new buildings erected across Germany possessed aspects of modern as well as ancient architecture, using simplicity to create a contemporary look but not show a disruption from the Latin past. Hence, “the Nazi building program was no revolution in style. To the contrary, buildings were to serve as symbols of secure old values and national pride and power.” Accordingly, the classical Altar of Pergamum inspired the Zeppelinfeld Stadium. Nero’s Domus Aurea influenced the Chancellery in Berlin and mosaics seen at Pompeii prompted the mosaic room inside of the Chancellery. The ancient stadium of Herodes Atticus inspired the monumental Deutche Stadion and the Kongresshalle traces its origins back to the Flavian Amphitheater, more commonly known as the Coliseum.

_{— Leda Wilkins Johnson: "Propaganda and Ideology: The Architecture of the Third Reich", Greensboro Historical Review, 2, PDF}

When not looking for stylistic comparisons, but eyewitness 'confessions', we see just as much emerging:

Their hope was that the buildings of Third Reich, when and if that Empire fell, would also express its lasting power. This macabre preoccupation helps to explain the dislike of reinforced concrete as a building material: both [Hitler/Speer] believed that it would appear undignified in ruins. Yet Speer nearly always used reinforced concrete, under limestone cladding, because it helped him build at the speed Hitler wanted. And it did look undignified in ruins.

These conversations about the ruins of ancient empires shed some light on the nature and development of Speer's beliefs about the relationship of architecture and politics. Many of Speer's buildings, insofar as they made reference to the past, appear distantly classicizing. Speer himself, in the first memoir that published after his release from Spandau, stated that the principle historic inspiration for his work was Greek architecture of Doric order-this was, he thought, the most noble of past architecture. In addition, he said, Hitler thought, and he himself believed at the time, that the Greeks were the ancestors of the Aryans; if Speer were to attempt a truly Germanic architecture, the Doric was the appropriate model.

There were many sources for this curious idea. Hitler did conflate the Greeks and the Aryans, as some archaeologists had already done early in the century. The association of Greece and German nationalism had long roots in German architecture, especially in Bavaria: it influenced, for example, the patronage of Ludwig I and the work of Leo von Klenze. Speer himself may have picked up the association not from Klenze, but from reading German literature of the Romantic period, which he liked. But Speer was also interested in archaeology. Like other German architects of his time, his training in architectural history was imparted mainly by archaeologists. Of his teachers in that field he especially admired Daniel Krencker, Roman archaeologist and excavator of the Imperial Palace at Trier, and Walter Andrae, assistant in German excavations at Babylon and himself the principal excavator of Assur.

WHAT? Wait a minute!

Assyria was an empire, and a mighty one at that. But it was a Semitic people who ran it! So, how did that creep in?

There is persuasive visual evidence that Andrae's reconstruction drawings of the main buildings at Assur, the Assyrian Empire, formed the most direct influence upon Speer's designs. Speer need not have known much ancient history to have realized that Assur was the center of a Semitic empire, and that the peoples who produced these buildings could not by any stretch of the imagination be supposed to have been Aryan or Indo-European. (The often used interchangeably, even by reputable ancient historians).

Yet in his Spandau Diaries, published in 1975 but supposedly written while he was still in prison, Speer admitted the possible importance of Assyrian models as influences on is own designs. How are we to explain this contradiction?

It is always wise to regard an architect's explanation of his work with a healthy mistrust, and this principle is even more useful in the case of a man like Speer, who had so many explanations to make.

Not lest we forget that Speer is an extremely unreliable source on the Third Reich in every aspect.

Most architects draw upon a variety of visual sources in a relatively unconscious way. When Speer saw Andrae's drawings he had not yet met Hitler or joined the Nazi Party; hence he had not yet learned to believe that architecture should have some ideological content. Probably he retained memories of Andrae's teaching images of an especially old, and newly discovered old, empire, which, by association, suited the idea of "ruin value" in architecture. Probably he did not think through the ideological implications of taking for his models the products of a Semitic people.

His protestations of admiration for Greek architecture, however, must have been conditioned by some notion of what he thought he ought to say, as a Nazi, and by a belief that this was what Hitler would like to hear.

For Hitler's sake, and, one must assume, for his own sake too, Speer was committed to finding some expressions for the nationalism of Nazi ideology, as well as for its references of populism. He explicitly rejected the "Germanic" styles of some Nazi architects. In the search for a rationale, it was the link between German, Aryan, and Greek which seemed to fit.

Clearly, though, Speer's overriding desire was to create an architecture which looked durable and old.

_{— Barbara Miller Lane: "Architects in Power: Politics and Ideology in the Work of Ernst May and Albert Speer",The Journal of Interdisciplinary History, Vol. 17, No. 1, The Evidence of Art: Images and Meaning in History, 1986, pp283–310.}

And despite all these caveats – especially regarding Speer as witness, for anything; and the eclectic tendency of architects; and the biggest of all: ruins of excavated foundations tell as much about the imaginative powers of an archaeologist doing a reconstruction as about 'the real thing' – the Andrae drawing of that temple is indeed conspicuous:

_{— Reconstruction drawing of the Temple of Tukulti-Ninurta at Assur, 1921, in Walter Andrae: "Der jüngere Ishtar Tempel in Assur", J.C. Hinrichs: Leipzig , 1935. PDF}

Note that the reconstruction drawing is based on findings from excavations like these:

(src)

With that site looking lime this today:

— Telul al-Aqr (ancient: Kar Tukulti Ninurta)

For comparison, during the Seleucid period, we now see such reconstructions:

— Uruk/Warka, situated in modern-day Iraq, is one of the first cities in the world and was populated almost without interruption for over 5,000 years – from the 4th millennium BCE to the 1st millennium CE.

During the Seleucid period (3rd – 2nd century BCE), Uruk was an important religious centre featuring large complexes of sacral architecture, such as the “Bit Resh” and its adjoining ziggurat, dedicated to the divine couple of Anu and Antum; the Irigal, home of the goddess Ishtar; the “Bit Akitu” (house of the New year’s festival); the Eanna ziggurat; and the Karaindash temple.
_{— Uruk Visualisation Project. The Seleucid period. Artefacts}

This 'influence' of Assyrian architecture as envisaged by German archaeologists may be felt to a degree in Speer's few buildings and many models. That he may have been one of the most important architects during Nazi times may be true, but not the only one, and not even the most prolific.

If we therefore reformulate the title question, with an emphasis on 'Speer' and add an 'also' 'for some projects' as well 'Assyrian' or 'Babylonian' into that sentence then we might see some influence of Assyrian architecture. For Speer, but not 'Nazi architecture'. Says Speer himself (have your salt ready):

There was no "Fuehrer style," for all that the party press expatiated on this subject. What was branded as the official architecture of the Reich was only the neoclassicism transmitted by [Paul Ludwig] Troost; it was multiplied, altered, exaggerated, and sometimes distorted to the point of ludicrousness. Hitler appreciated the permanent qualities of the classical style all the more because he thought he had found certain points of relationship between the Dorians and his own Germanic world.
[…]
It surprised me that France also favored neoclassicism for her public buildings. It has often been asserted that this style is characteristic of the architecture of totalitarian states. That is not at all true. Rather, it was characteristic of the era and left its impress upon Washington, London, and Paris as well as Rome, Moscow, and our plans for Berlin.
[…]
The design for Goering’s headquarters provided for extensive series of stairways, halls, and salons which took up more room than the offices themselves. The heart of the building was to be an imposing hall with a great flight of stairs rising through four stories, which would never have been used since everyone would of course have taken the elevator. The whole thing was pure spectacle. This was a decisive step in my personal development from the neoclassicism I had first espoused, and which was perhaps still to be seen in the new Chancellery, to a blatant nouveau riche architecture of prestige. An entry for May 5, 1941, in my office journal records that the Reich Marshal was highly pleased with the model of his building.
_{— Albert Speer: "Inside the Third Reich", Macmillan, 1997. Not to be read unless consulting Gitta Sereny: "Albert Speer. His Battle with Truth", PanMacmillan, 1997. And: Paul B. Jaskot: "Anti-Semitic Policy in Albert Speer's Plans for the Rebuilding of Berlin", The Art Bulletin, Vol. 78, No. 4, 1996, pp.622–632.}

And looking at one of Troost's concepts for a 'Soldier's Hall', we see almost a copy of the Pergamon Altar, now in Berlin, from Greek Anatolia, only pumped up to enormous size:

— Michael Fröhlich: "Die Soldatenhalle in Hitlers Neugestaltungsplänen der Reichshauptstadt", 2018.

But that was then clearly not the only influence visible in 'his style', which indeed is exemplary with other styles in the Neue Reich Chancellory: reduced neo-classic towards the street (hints at pilasters), with added modernist elements (style of entrance, windows; somewhat 'transparent'/low set, in contrast to the oversizing of all of it); while the back is even more different from the front, real colonnades with real capitals (of course with squarish instead of proper Doric style):

Other examples of Nazi architecture are Wewelsburg, an originally renaissance castle rebuild to Himmler's SS-taste, or Napola's, NS schools, like Ballenstedt: (alt src, more pics)

Or the decidedly modernist airport Berlin Tempelhof, designed by Sagebiel (subordinate to a Jewish boss previously) but completed by Speer:

It might also be noteworthy that if the anti-Semites would have been aware of these connections between Assyrian style and some of Speer's ideas, then for sure they also would have been referring this peculiar specialty to ancient bible stories of both Assyrians (later than the one depicted) and Babylonians crushing Israelite kingdoms and destroying them. There is no positive relationship discernible in using any recognisable 'Jewish' architecture as a model or inspiration in any 'Nazi architecture'. If you, dear reader, see a logical gap there, you're right. It is called racism as ideology.