Paper Mario and Strong Bad's Cool Game for Attractive People are my comedy kings of late. Most of my other funny moments aren't from games actively trying to be funny (the two technicians assembling the rocket launcher in Perfect Dark's third mission for example, one of whom has an amusing Liverpool accent).
'Spikey Joe and Mustache! Together they fight crime! Yeah... that'd be a killer TV show!'
-Spikey Joe, proposing a lucrative deal with Mario
'It's not like the secret entrance is just going to be labelled 'Secret Entrance'. We have to find some way of revealing it.' -Punio, before later finding the way to reveal the entrance, which is indeed labelled 'SECRET ENTRANCE'.
Strong Bad: What'd you bring that ugly misshapen stick for?
Homestar: Oh, she organized the protest rally!
SB: Not Marzipan! Though that's a pretty good one... gotta remember that!
Coach Z: Oh, that's our orffigy of the King of Town. Um, arffigy! RPG! First-Person-Shooter!
Strong Bad: This is never gonna work! I'm gonna need a translator. The Cheat, ask Genghis Can't here if he wants to join Strongbadia.
The Cheat: *The Cheat Noises*
Strong Mad: THE CHEAT'S MY BEST FRIEND!
The Cheat: *The Cheat Noises*
SB: Ohhhh, so you're his best friennnnnd, it's all starting to make sennnnnse.