This is a hard one for me to answer. Not because I can't think of any games that are funny, intentional or not, but because I can think of too many. x) Not a bad problem to have!
The House of the Dead 2 above made me immediately think of this game. They obviously knew that the series was infamous for it's voice acting and while they could have tried to make it bad on purpose, in my experience that usually doesn't work out too well. So I'm glad they took the direction they did instead. I remember it being a solid on-rails shooter game too!
This one's a classic. I don't think it's particularly fun to play and some of the referential humor may fall flat depending on age or origin (do kids these days know what A Clockwork Orange or the Matrix are?), but it's still a hoot throughout.
I've not really played the prior Hitman games except a couple of demos, but I loved the World of Assassination trilogy. It is chock full of intentional and unintentional comedy. Agent 47 is such an incredible dork too. Like 80% of what he says is just some variation of "tee hee I kill people!" And hey, if you're a creatively bankrupt streamer, after you finish overreacting to Amnesia: The Dark Descent you can always dress up in the funny clown outfit and throw silly things at people for free clicks!
Another classic. The beginning can be a slog and it's not the best playing game I've played, but it succeeds in basically every other way. Great writing, a fun cast of weird characters, good art design and an overall interesting concept. You can pick it up for cheap on Steam or GoG, but it's also out for the Xbox and PS2, though I hear those versions are not as good as the PC one.
