I laugh at a LOT of places in these movies. I'm going to break this down by movie and try to list just two for each so this post doesn't go on for pages and pages :-)
I tend to laugh over things that delight me, surprise me in a good way, unexpected-but-perfect word choices (mostly a book thing), and characters behaving in a way that sort of typifies exactly who they are and what they're like. And complete absurdity, that'll set me off too.
The Fellowship of the Ring
Pippin's smart-alecky face when he says, "We've had one yes. But what about second breakfast?" He's so confident that Aragorn will be like, "Oh! Yes, I forgot about second breakfast. By all means, let's stop, build a fire, and cook another meal. My bad."
When the Council of Elrond has agreed that Frodo will take the Ring into Mordor, Sam jumps out of hiding and says he's going too, and Elrond is like, "Aren't you sweet." But then Merry and Pippin pop out too and run past him, and Elrond has this expression of, "Gah! My city is overrun by these Hobbits!" Cracks me up.
The Two Towers
Gimli's grousing about being a natural sprinter, when he and Legolas and Strider are doing their awe-inspiring run to the rescue. Really never gets old. It's totally just there to be funny, and I usually don't like jokes-for-the-sake-of-jokes, but that one does amuse me.
Merry and Pippin's expressions at various times during their time with Treebeard also make me laugh, but not always the same ones -- guess it depends on my mood.
The Return of the King
Merry and Pippin and Gimli again, this time when everyone arrives at Isengard and get greeted by two jolly hobbits. That whole section cracks me up.
The drinking contest between Gimli and Legolas is also quite amusing, and I definitely laugh when Legolas says, "I think it's beginning to affect me," and then Gimli falls over.
An Unexpected Journey
Bombur catching an egg in his mouth, and everyone's reaction to it. Especially since I know now that that was the first take, and no one expected him to be able to do it! I mostly dislike the whole dwarves-invade-Bag-End-and-mess-everything-up sequence, but that part is very funny.
I laugh at all the dwarves splashing around in the fountain at Rivendell in the extended version. Well, mostly I laugh at how discomfited the elves are by this -- I sympathize, but at the same time, it's hilarious.
The Desolation of Smaug
Kili telling Tauriel a scary story about how his rune stone will curse anyone who's not a dwarf if they look at it. He's such a teenage boy there, and I love her reaction -- this look of, "Oooookay, let's just walk away from the crazy dwarf, then."
When all the dwarves are coming up out of the toilet into Bard's house, and little Tilda asks, "Will they bring us luck?" I laugh because it's a perfect little-kid thing to ask, and also because yes, they will. Thanks to everything that goes down once those dwarves reach the mountain, your Da is going to be King of Dale in a couple years, kiddo. Lucky dwarves, indeed.
The Battle of the Five Armies
When Thranduil asks Bilbo if he's the one who helped the dwarves escape, and Bilbo does his nose-twitch thing and says, "Errrrrrrrrrrrrrrrrrrrrrrrrrrrm, yeeeeeeeeeeeeeeeeeeeeeeeees." Who could help laughing there?
Alfrid readjusting the gold stashed in his dress while he's preparing to scuttle off for good. I laugh because it makes me think of Little John in Disney's animated Robin Hood, also buxom with stolen gold. Also, Alfrid's trying to be so smart, but he looks ridiculous, and he's trying hard not to acknowledge that he looks ridiculous, and so, yup, it's funny.