Besides Arab/Muslim countries re Israel, are there other countries which don't recognize passports of any other countries?

Having some passport not recognized by some country is not that unusual. It’s pretty much certain to happen somewhere for passports of countries with partial recognition.

Someone else mentioned passports of Northern Cyprus (which is only recognized by Turkey), which according to Wikipedia, is only accepted in 6 countries. But there are many other countries in the same boat.

The Republic of China passport: The Republic of China is recognized by 21 UN members plus the Holy See. But its passport is accepted in almost all countries. However, there are some peculiarities. According to Wikipedia, a few countries (Argentina, Kenya, Laos, Nepal, Serbia) will issue the visa on a separate sheet, not on the passport itself, presumably due to not recognizing the passport. Brazil apparently will require the person apply for a Brazilian laissez-passer as the document to put the visa on. China (People’s Republic of) will issue some kind of permit instead.

The page for Kosovar passport does not explicitly mention any country where it is not accepted. But common sense would dictate that it is not recognized by Serbia.

The Nagorno-Karabakh passport is not recognized anywhere, according to Wikipedia (though it’s unclear if that means it cannot be used anywhere). For Transnistrian passport it says “not valid for travel to most countries” (not really sure what that means). The pages for Abkhazian passport and South Ossetian passport do not explicitly mention any country where they are not accepted. But again, they would be at least not recognized by the countries that their territory is claimed by (Azerbaijan for Nagorno-Karabakh, Moldova for Transnistria, Georgia for Abkhazia and South Ossetia). The page for Palestinian Authority passport also does not mention where it is not recognized; but for that case it is conceivable that it could be recognized everywhere. For Somaliland passport, it says it is accepted in 8 countries as unofficial travel documents, which probably means it is not accepted mostly everywhere else.

South Koreans visiting North Korea will be issued a visa on a separate piece of paper.

People’s Republic of China doesn’t recognize Taiwan passports (Republic of China passports). Taiwan in turn doesn’t recognize passports issued by PRC.

See Wikipedia: Mainland Travel Permit for Taiwan Residents

There is the Principality of Huttriver, which isn’t recognized by the Australian authorities, yet its prince seems to travel on a Hutt passport:

BRENDAN HUTCHENS: The Prince still travels abroad on his Hutt River passport, as do the principality’s citizens, who have many stories to tell from passport control.

Source

One case I know is Northern Cyprus. In the wake of the 1974 invasion, Turkey created a new state that still administers a part of the island. That state is recognized (and strongly influenced) by Turkey. Its situation (generally unrecognized but accepted by another country) is therefore in a way the “reverse” of Israel’s situation (widely recognized but opposed by some countries).

Northern Cyprus issues passports that, a few years back could be used to travel to at least a few other countries (you needed to go through Turkey however as direct flights to other destinations were basically non-existent and crossing to the South to get a flight there was impossible). Where things get a little “funny” is that those countries would typically avoid putting any stamp in the Northern Cypriot passport, instead delivering visas on a separate piece of paper as a way to signal their refusal to recognize the Turkish Cypriot state. I have a friend who could study in the US on such a passport.

Things have become much easier as part of Cyprus’s accession to the EU (even if the final resolution of the conflict that was hoped for did not happen). Since then, the border between both parts (the “green line”) is open and people from Northern Cyprus can go to the South and get a Cypriot passport (which is of course much more convenient, widely recognized with visa-free travel and immigration in the EU and more).

My understanding is that you need to have some pre-1974 roots in the island to get a passport from the Republic of Cyprus. The conflict is old enough for many people to have been born after the division but if they can present their parents’ birth certificates and the like, Turkish-speaking residents of the Northern part can get such a passport (not sure about the exact rules or practicalities but I know some who have).

I am not sure of the status of the people who came to Cyprus from Turkey after 1974. I suspect some of them might be able to get Turkish papers.