Years after Big Bird's run for president, Oscar showed Cookie Monster a crumpled-up copy of Big Bird's presidential poster. This inspired Cookie to run himself, and sing the immortal "If me were president Cookie." It was a good song that might never be heard again. :) Oscar accompanied Cookie Monster to various speeches in front of an anything muppet audience. The one quote from Cookie I recall: "Give me liberty, or give me cookie." In the end, both Cookie and Big Bird dropped out of the race for the same reason: It's really hard to run for president.
Years later, Oscar would run for mayor of New York City. His campaign ad began with a voice-over from Telly Monster:
TELLY: Oscar, do you think this city is a mess?
OSCAR: Yes. This city is a mess. But it's not enough of a mess. Vote for me and the Big Apple will become the rotten apple!
Again, Oscar dropped out of the race within one episode, I think mostly because of all the good things he realized he would have to do.
Some other notable fictional folk who have run for president: Snoopy (In a promotion with Dolly Madison, I think), Winnie the Pooh (in a promotion with Sears, I think) , Olive Oyl (in a dream sequence from a cartoon) .