1. Children's game similar to
musical chairs, in which an object (traditionally a
potato, now usually played with something softer) is passed/tossed/shoved around a
circle, and when a caller says "Stop," whoever has the object is out of the game.
2. Expression meaning something uncomfortable forced on a person, which they would rather transfer to someone else as soon as possible.
3. Pop song (by the group Pugilism and the Third Autistic Cuckoo) in Douglas Adams' book The Long Dark Teatime of the Soul in which the lyrics come from an overheard conversation about a contract which is a hot potato in sense #2.
4. 1984 NBC game show.