The opening ceremonies had pickup trucks enter the stadium while people spelled out "How Y'all Doin?" Seriously.
One gymnast on the Chinese men's team in 1996 paused part way through his routine, to wave hi to the audience using his foot. Doubly funny given the reputation the Chinese have for being very straitlaced and serious.
In the 1000 m short track speed skating, a big crash led to all the athletes falling except for Steve Bradbury of Australia, who was too far behind everyone else. Because of this Bradbury, who was only in the finals because of another crash in the semis, wound up winning the gold! Brabdury has since become part of the Australian dictionary for "a fortuitous event".
2008 Summer Games — Beijing, China
The Brazilian airing of the 2008 Olympics' opening ceremony was unintentionally hilarious, due to the added commentary by Oscar Schmidt, pretty much the best basketball player Brazil's ever had, who spent the ENTIRE CEREMONY complaining about how he regretted refusing to carry the Brazilian flag when he was offered (in 2004) because he needed to rest for the game the following day. The game that, as he constantly reminded viewers, nobody remembered whether or not they'd won. NOT EVEN HIM.
Galvăo Bueno, narrator of that ceremony and most sports in that network, also makes most of his transmission unintentionally hilarious (and/or obnoxious) due to his tendency of obviousness ("went to the ground and fell!") and trying to show his smartness ("without omelette, nobody does eggs", "his disposition is very high, he looks like a caged lion").
At the closing ceremony — London's 8 minutes. Boris Johnson looking gormless while waving a flag, funny. That was just the start of eight minutes of simultaneous mass charm, hilarity, weirdness and cheese. Words can't describe it... Just watch it.
For that matter, David Beckham kicking off a football in said closing ceremony - and hitting a guy on the head.
Later that day, Mr. Johnson again, after stating that the British had invented the East Asian dominated spot of ping-pong with a game called "wiff-waff":
Wiff-waff is coming home!
There was much speculation as to who the music performance would be by — The Doors, Queen, etc. On the big night, the commentary went like this:
Canadian ice dancing gold medalists Virtue and Moir's funny showcase performance, which appeared to be a homage to teen movies (he was a(n ice) Sk8erBoi, she did ballet...)
The closing ceremony, where in addition to the usual impressive ceremony there was a menagerie of humorous Canadian stereotypes ranging from mounties, beavers, lumberjacks (wearing hastily-added gold medals!), and hockey players to oddly sexy woman dressed as giant maple leaves.
A number of Canadian comedians expressing the country's appreciation towards their guests, including Michael J. Fox heartwarmingly declaring everybody as honorary Canadians - which he then claims means that Canada won every gold medal of the games.
William Shatner's speech — really, that entire closing ceremony was a cavalcade of hilarious self-parody.
2012 Summer Games — London, England
Most of Danny Boyle's opening ceremony, which was called "wacky", "off-the-wall", and "bonkers" in many quarters.
Rowan Atkinson's quasi-Mr. Bean segment. The Chariots of Fire theme starts with all the drama and seriousness one would expect from the situation... and then the camera pans over to him sitting at the keyboard, clearly bored that his part only involves hitting the same "pulse" note over and over. He manages to take out his iPhone to take a picture, sneeze and play the keyboard with an umbrella so he can get a tissue out of his backpack, and then falls asleep, dreaming himself into the movie's "running on the beach" opening sequence. After cheating to win the footrace (he takes a cab!), he wakes up to find the rest of the orchestra has finished the song, and everyone is staring at him. The conductor (highly respected conductor Sir Simon Rattle!) scrunches a hilariously displeased expression and signals for the final notes. So he does so, only to hit one messed-up note at the end.
The Queen parachuting out a helicopter with James Bond. And American viewers got the bonus of Meredith Vieira and Matt Lauer apparently convinced it was real!
The Queen's corgis being filmed in the Shaky Cam method of the modern Bond films.
Daniel Craig looking absolutely bemused at the Churchill statue waving at him, especially the part where he follows the Queen after being escorted to her office.
There's something just so cracktactic about Mary Poppins defeating a giant Voldemort... especially since if the music was any indication we were supposed to take it dead seriously. British humor is like that. They're sure, but you're never quite sure...