In the first game, some of the commentary from the Delta operators, particularly Holiday, is hilarious.
Holiday: (disarming explosives) Man, whoever rigged this was a pro. Bishop: Is that supposed to make me feel better? Holiday: He may be a pro, but I'm the master.
Betters has plenty of commentary to add over the course of the mission, like when dealing with Alice Wade:
Betters:Those soldiers are after you. If they capture you again, they're going to interrogate you - and believe me, their methods are very unpleasant. Alice Wade: I'll take my chances. Betters: Stubborn bitch, ain't she?
Later, when Jin and the Point Man meet back up:
Jin: Thank God you're alright! Betters: You two can make out later, now get on the chopper!
Alice freaking out if the Point Man decides to leave the elevator to take the fight to the Replica.
Alice: You don't have to kill everybody!
Really, the entire escort sequence. Between the oddly cheery music in the elevator to Alice's increasingly horrified reactions to the Point Man getting his Blood Knight on, its one of the most hilarious moments in the game.
When it comes to unscripted moments, smacking an enemy while in Slow-Mo will produce a hilarious bobblehead effect. You can almost imagine a Looney Tunes-esque boing noise whenever their heads start bouncing back and forth.
In Project Origin, Jankowski's reactions if you screw around in the plaza.
Griffin: What kept you? Jankowski: Bucket had to take a bath in the fountain.
Everything about Snake Fist.
Stokes: Who the hell are you? Snake Fist: You can call me...Snake Fist. Stokes: (incredulous) ...are you fucking kidding me?!
Snake Fist: Alma covets you. She's a....coveter. You're like free pizza at an anime convention. She can smell you. And she wants to consume you.
Keegan and Jankowski get sadly little banter, but what is there is hilarious.
Jankowski: A lot of blood. Looks like he got hit in the aorta. Keegan: The aorta? Maybe he got hit in the femoral, or subclavian? Jankowski: Dude, making fun of someone's intelligence is a sign of immaturity.
A few moments later, after the first encounter with an Armacham goon:
Jankowski: Keegan, you okay? Keegan: I think they hit my aorta! Jankowski: Dude, give it a rest.
Hiding behind the cardboard donkey in Wade Elementary's auditorium causes the enemy soldiers to utter this hilarious line.
Armacham Troops: The ass! He's behind the ass! Shoot the ass!
The linked picture gives a perfect example of the primary source of comedy in these games: the Ragdoll Physics are just amazing, from enemies flinging themselves over guardrails to getting limbs stuck in walls.