On the Southern Highlands, after defeating a cultist and the khazra ghosts he summoned, the player finds his journal:
Yes, with these heathens at my command, the kingdoms of the world shall bow to the might of... THE LORD OF GOATS!
...I may have to work on the title a bit more.
In the first act, it is possible to stumble on a "Forlorn Farm". Upon defeating the enemies there, you are invited into the cellar where a farmer and his wife hid. If you go down, you will find that the wife is long dead, nothing but a desiccated corpse in a rocking chair whose head proceeds to fall off; the farmer may or may not be oblivious to this - each line from him is part of a Hurricane of Puns, much to the dismay of the Player Character.
Farmer: [after the corpse's head topples off onto the floor] Oh, she's nodded off.
Farmer: My wife can't help with the planting and harvesting. She's nothing but skin and bones!
Farmer: It can be dangerous in the wilderness. Being out this far scares my wife to death!
Farmer: You wouldn't know it, but this basement gets right chilly at night. Cold as the grave!
Farmer: You should have been there when I met my wife. I lost my heart and she lost her head.
PC: You must stop this now.
Farmer: I can't take all the credit. My wife handles the business side of things. She has a good head on her shoulders!
PC: Hmm... Really?
During Act III while a demon ambush is going on:
Captain Haile: Private, if you need to be reminded which end of the sword goes where, you haven't been paying attention!
Captain Haile: You there! Damotrius! Stop moping about! If this was your funeral I'm bloody well certain you'd have been told!
And when you're battling demons with him in the Blaze of Glory event at Rakkis Crossing:
Captain Haile: Those worms have come up from behind! Bloody adorable! They're trying to ambush us!
Also from Act III, when raising the last catapult from its moorings, your sole NPC aid comes from a grizzled lieutenant whose vinegar-laced "encouragement" straddles the line between hilarious and Crazy Awesome:
Lieutenant Clyfton: I'm sorry, catapult, I'm not strong enough to lift you on my own. Would you mind convincing my friend over there to lend a bloody hand?!
Made even funnier after you finally get the damn thing lifted, and he becomes amazingly sheepish:
Lieutenant Clyfton: Done! I...uh...promised Captain Haile I'd watch my temper. Do me a favor and don't tell him that I shouted at you a little, all right?
In Act I after rescuing Cain, you can stumble upon a conversation between a villager and a tired patron in New Tristram:
Villager: You know what would take everyone's mind off their troubles? One of Theodas's great comedies.
Tired Patron: I've seen one of them. Two men dressed as women, screeching at each other and tearing their bodices? Oh, that'd cheer me right up.
Villager: Glad to hear it. Rehearsal is in three hours!
During Act III in the base area:
Little Boy: Stab those bloody demons in the arse!
In Act II, during the quest to collect the various parts of Zoltun Kulle:
Lyndon: People say I'm bad, but no one's ever had to imprison me and dismantle my body parts. A little context would be nice.
Barbarian: Yes. My people were the guardians of the Worldstone.
When the old soldier and his wife at Bastion's Keep in Act III aren't being heartwarming, they're usually being funny instead. In particular, after one event they reminisce together about the time she was kidnapped by barbarians: the soldier describes how he was beside himself until they returned her with an apology, and his wife comments that their leader still sends her a bundle of hides every year and is the nicest kidnapper she's met.
Kormac the Templar's truly over-the-top reactions upon encountering Elite Mooks. Made funnier when made against elites in Whimsyshire, the land of unicorns, cuddly bears, flowers and happy clouds.
In the Caldeum Bazaar in Act 2, there are two kids (a boy and a girl) running around with their pet rabbit. They eventually make their way to an Iron Wolf in the center of the map, where the man kills the rabbit with his sword, splattering blood all over the ground. The boy and the girl then run in opposite directions, crying.
Taking Lyndon through the bulk of Act 2 or 3. His endless complaints are rather amusing...helped greatly by his voice actor's excellent work.