What happened to Hitler's personal possessions and real estate?

Hitler's last will and testament gave "items of sentimental value" to his remaining family, his art to a gallery in his home town, and any wealth to "the National Socialist Party if it survives, or the German state if it does not". He explicitly said he didn't care what happened to it if neither of those organizations existed.

Specifics about what ended up where are harder to come by. According to Wikipedia, his home in Obersalzburg was destroyed and looted near the end of the war, and the ruins were demolished in 1951. A golf course is there now.

Hitler's other residence, the Wolf's Lair, was destroyed by the SS as the Soviets approached in 1945. The ruins are still there in Poland, and are a bit of a tourist attraction.

The German's attempted to demolish the Eagle's Nest in 1945, but fled before they finished. It was captured by the US. Post-war, it was used as the headquarters for Organisation Geheln, a US-German intelligence agency that eventually became the Federal Intelligence Service. In 1991, it was sold back to the family the Nazis appropriated it from. It's an apartment complex now.

One problem I had researching this is deciding what was Hitler's and what was the state's. He didn't seem to see much distinction.

Just adding to the other answers, Hitler's chief aid Julius Schaub was ordered to destroy his boss' personal possessions.

Schaub was instructed to burn all of Hitler's personal belongings and papers in the garden of the Reich Chancellery.1 Schaub then flew to Munich and did the same in Hitler's private apartment at the Prinzregentenplatz and then at the Berghof on the Obersalzberg.1 His final act as aide and adjutant was to destroy Hitler's personal train, the Führersonderzug.

Some things were taken by Nazi officials and ended up in Argentina (according to some Argentine investigators quoted in a Daily Telegraph report). These include a magnifying glass used by Hitler (there are photos showing him using it).

Some things were taken or looted by the Americans (and probably the British also). Hitler's 1939 Mercedes was seized by the US army. There's a list of other things in an article Most Expensive Adolf Hitler Memorabilia Sold at Auction (but not all of these were Hitler's).

Short answer:
Hitler's will, as such, probably played no role in the redistribution of his personal property.

Reliable sources could, no doubt, be found to confirm that the Allied intension from the beginning was to confiscate the property of the NSDAP and it's main party leaders.

Careful reading of the Wikipedia sources show that, when no longer needed by Allied Forces, property was transferred to the local state governments (no doubt with conditions).

The Wikipedia (German only) Führersperrgebiet Obersalzberg article shows that this held true also for personal property

Die Schallplattensammlung Adolf Hitlers gelangte in das amerikanische Nationalarchiv.

Adolf Hitler's record collection reached the American National Archives.

which was not necessarily allways returned.

Adolf Hitler's Munich apartment now belongs to the Bavarian state.

Based on this German article from 2013

Eingezogen war Hitler am 1. Oktober 1929 samt Personal. Eva Braun kam oft zu Besuch. Und auch allerhand Größen der damaligen Zeit waren am Prinzregentenplatz zu Gast. Ende 1938 ging − wohl mit Hilfe der Partei und dank der sprudelnden Einnahmen aus dem Verkauf seines Buchs "Mein Kampf" − das ganze Haus in Hitlers Privatbesitz über. Bis zu seinem Tod 1945 blieb er mit dieser Adresse in München gemeldet – auch wenn er nach Ausbruch des Kriegs kaum mehr in der Wohnung lebte. Dabei war gleich ab 1939 für ihn ein Luftschutzbunker mit einer 4,75 Tonnen schweren Stahldecke und Eichenholz-getäfeltem Führer-Raum gebaut worden.

Hitler was moved in on the 1st of October 1929, including personnel. Eva Braun often came to visit. And also all sorts of notable people of the time were guests at Prinzregentenplatz. At the end of 1938 - probably with the help of the party and thanks to the bubbling revenues from the sale of his book "Mein Kampf" - the whole house passed into Hitler's private ownership. Until his death in 1945, he remained registered at this address in Munich - despite the fact that he he hardly lived in the apartment after the outbreak of war. In 1939, an air raid shelter with a 4.75-ton steel ceiling and oak-paneled "Führer room" had been built for him.

Traditionally, with the exception of the present Chancellor, a Chancellor used the provided flat within the Chancellery.

This residence was situated in the (Old) Reichskanzlei, Wilhelmstraße No. 77 (former "Palais Schulenburg").

• to my knowledge, Hitler had no property in Berlin, possibly none outside of Bavaria.

Hitlers privates Eigentum fiel an an den Freistaat – darunter auch das gesamte Gebäude am Prinzregentenplatz. Seit 1949 residieren hier verschiedene Polizeibehörden und verhindern, dass die Wohnung eine Wallfahrtsstätte für die Ewiggestrigen wird. Heute ist der Prinzregentenplatz 16 Sitz der Polizeiinspektion 22.

Hitler's private property fell to the Free State [of Bavaria] - including the entire building at Prinzregentenplatz. Since 1949, various police authorities have been residing there and preventing the apartment from becoming a place of pilgrimage for the die-hard. Today the Prinzregentenplatz 16 is the seat of the police inspectorate 22.

In 1933 a property was bought in Obersalzberg, which was renamed to Berghof in 1935/36 and served as a Sommer residence.

In 1937 the building of the Kehlsteinhaus (known as the Eagle's Nest in English-speaking countries) was commissioned by the NSDAP and was not considered to be private property.