I was inspired to ask this question after reading about the 1980 and 1984 Summer Olympics boycotts. I noticed that both Iran and Albania boycotted both games, apparently for political reasons on each occasion.
Iran boycotting both games makes some logical sense, as I imagine they were not keen on either the Soviet invasion of Afghanistan or America's foreign policy in general. As for Albania, I really have no idea...
Were there any other countries that fit this description?
I'm not certain if this exactly answers your question, but the People's Republic of China and the Soviet Union had a difficult relationship for much of the Cold War while, at the same time, both were adversaries of the United States
China and the USSR came to blows, for example, in 1969, with some loss of life on both sides. And while not really related directly to your question, following the end of the Vietnam War, China and Vietnam, which was somewhat more in the Soviet orbit, fought a border war.
Yugoslavia also had somewhat difficult relationships with the Soviet Union, particularly during the Stalin era. Yugoslavia was expelled from the Cominform for a time. While Yugoslavia was a recipient of Western aid via the Marshall Plan, it never became really friendly to the West either.