Are there any countries that hosted the Olympics that no longer exist?

The former Yugoslavia hosted the 1984 Winter games in Sarajevo. Yugoslavia has ceased to exist since 1992.

https://en.wikipedia.org/wiki/1984_Winter_Olympics

https://en.wikipedia.org/wiki/Yugoslavia

Countries that hosted the Olympics that no longer exist:

Summer Olympics - West Germany in 1972 and the Soviet Union in 1980
Winter Olympics - Yugoslavia in 1984

https://en.wikipedia.org/wiki/Summer_Olympic_Games#List_of_Summer_Olympic_Games
https://en.wikipedia.org/wiki/Winter_Olympic_Games#List_of_Winter_Olympic_Games

A previous answer said that West Germany, host of the 1972 Summer Olympics, doesn't exist anymore. However, it was the Federal Republic of Germany that hosted the game, and that country very much exists today. In the early 90s, it absorbed the states that used to make up the German Democratic Republic. (The GDR doesn't exist anymore, but it never hosted the Olympic Games.)

It would be more correct to say that the Germany that hosted the 1936 Summer (and Winter) Olympics no longer exists.

(And the previous answers of Yugoslavia are correct, of course.)

It's an interesting point that akg1486 raises which seems to hinge on the what the term "ceases to exist" should mean in this context. In Olympic terms (which is what we are talking about after all) there were 2 countries which have now become one. I would say that means that 2 countries no longer exist in Olympic terms and have been replaced by a new one. Countries can "no longer exist" through amalgamation just as they can no longer exist through breaking up.

The united Germany is the enlarged continuation of the Federal Republic and not a successor state. As such, the Federal Republic of Germany retained all its memberships in international organizations including the European Community (later the European Union) and NATO, while relinquishing membership in the Warsaw Pact and other international organizations to which only East Germany belonged.

https://en.wikipedia.org/wiki/German_reunification

I came back to this answer because none of the answers here are completely correct, based on the discussion above. It seems clear that current Germany is the same legal entity as West Germany was in 1972, because what officially happened was that the East German states, which were always recognized under the German constitution as parts of the Federal Republic of Germany that could elect to return, were finally allowed to join it (and in return, the Federal Republic amended that constitution to accept these new borders as final and end its claims on the various Laender constituting Upper and Lower Silesia and West and East Prussia, as well as parts of Pomerania and Brandenberg, all of which were gobbled up by Poland and the Soviet Union). Conversely, as stated above by akg1486, it seems that 1936 Germany (the German Reich, or Nazi Germany) was a separate entity than the 1972 Federal Republic of Germany.

Thus, taking all of this into account, the countries that have hosted the Olympics that no longer exist are the following:

Summer Olympics: German Reich (1936), USSR (1980);
Winter Olympics: German Reich (1936), Yugoslavia (1984).

Using the IOC's country codes, there are three host countries which no longer exist. Their website Olympics.org lists the medals won by those former countries under their defunct IOC country code:

URS - Soviet Union (1952 to 1988) hosted 1980 Summer Olympics
YUG - Yugoslavia (1924 to 2000) hosted the 1984 Winter Olympics
FRG - West Germany (1952 to 1988) hosted 1972 Summer Olympics
https://www.olympic.org/ussr
https://www.olympic.org/yugoslavia
https://www.olympic.org/federal-republic-of-germany

GDR - Easter Germany - attended the Olympics 1968 to 1988
GER - Germany - attended 1896 to 1936, 1952 and 1992 -
https://www.olympic.org/german-democratic-republic
https://www.olympic.org/germany