The most memed one (at least stateside) has to be (CW: homophobic slur) this baseball one. Not really catastrophic for anyone besides him, though. The Diamond Joe "this is a big fucking deal" incident is more well known but it doesn't come with a copypasta-ready apology and also didn't really matter.

edit: scrolling through the hot mic Wikipedia article and there are a few fun ones in there:

On 14 August 2008, actor Ernest Borgnine was interviewed on Fox News when he was asked about the secret to his longevity. Laughingly Borgnine responded "I don't dare tell you," but then leaned over to whisper into the ear of his interviewer, but the whisper was caught by the microphone; "I masturbate a lot."

On 2 March 2009, footage of Dutch news anchor Eva Jinek asking whether to loosen another button on her shirt (which might reveal too much of her cleavage) right before a broadcast, while muttering and shouting mild curses in both English and Dutch, was accidentally leaked to and spread on the Internet. She exclaims "Yeah, boobies!" at one point, continuing with, "Mother of God. Yeah, if you have it, flaunt it!" and citing her mother who might tell her: "I can see your breasts!" She responded the next day, saying she was not annoyed nor embarrassed by the leaked video.

In 2005, French President Jacques Chirac was heard criticizing British cuisine to Russian President Vladimir Putin and German Chancellor Gerhard Schroeder, saying "The only thing that they have ever done for European agriculture is mad cow disease" and "You can't trust people who cook as badly as that." He was also heard criticizing haggis, saying "that's where our problems with NATO come from", and that British food was the second worst "after Finland".