To be honest, too many to list. Pretty much safe to say that Kirk striding around hoping that he's got the distant past nailed is one long howl of laughter.
The entire film is one of these: they go back in time to steal whales!
And it's McCoy who puts it explicitly out there:
McCoy: This is insane! You are proposing that we go backwards in time, pick up two humpback whales, bring them forwards in time, drop 'em off and hope they tell this probe what to go do with itself!
Kirk: That's the plan.
McCoy: Well that's crazy!
Kirk: If you got a better idea, now's the time.
"YOU POMPOUS ASS!"
Fridge Funny: When the Probe shuts down Spacedock, you can see the Excelsior in the hangar. That's right Starfleet, your shiny new ubership just conked out on you.... again.
Or maybe they simply haven't fixed it yet, given that the events of Star Trek III happened only three months ago.
The two garbage men, after witnessing the Bounty's landing in the park.
Garbage man A: Did you see that?
Garbage man B: No, and neither did you, so shut up.
Then seconds later:
Kirk: Everybody, remember where we parked.
Spock meets punk. Punk refuses to turn down ghettoblaster. Spock nerve-pinches punk. Everyone - the bus passengers, the audience in the theater, aliens watching this broadcast on satellite TV from 500 light years away - cheers.
And then Spock continues his conversation with Kirk as if he's still trying to speak over the punk's music, with Kirk's expression just screaming 'On top of everything else, I just don't have the energy to tell you you don't need to shout anymore.'
Let's not forget when they first tried to get onto a bus. They walked up the steps, there's a pause, and then they climb off again. The entire structure of the scene just makes the moment:
Spock: What does he mean "exact change"?
Kirk's reaction when he sees Spock in the aquarium tank mind-melding with the whales is priceless.
Any time Spock tries to swear.
"They like you very much, but they are not the hell your whales."
Gillian: I suppose they told you that too, huh?
Spock: The hell they did!
"One damn minute, Admiral."
Angry Driver: "Why don't you watch where you're going, you dumbass!"
Or possibly a commentary on how history sometimes works. Shakespeare's works, for instance, was aimed far more "low class" than people realize. Considering that Spock was almost incapable of sarcasm at this point in his recovery, they may have saying those authors ARE considered giants in the 23rd century, which is also funny.
Gillian Taylor, much like Edith Keeler, is not your typical 20th century human. Her facial expressions across the movie show she's not really buying what Kirk and Spock are trying to tell her, and she suspects something is up with them. The funniness comes when later, she's told the truth of things by Kirk (who knew she might not believe anyway), she was completely unsurprised. And when she sees the invisible Mr. Scott helping the vanishing materials, bumps headfirst into the invisible Bird-of-Prey, she's completely dumbfounded by all this, but recovers quickly. The actress's acting is what sells it.
The entire scene where they spring Chekov from the hospital is funny. From McCoy's disdain for 20th-century medicine, to the elderly woman who grows a new kidney, to Chekov's cheeky promotion of himself to admiral, to the way they get into Chekov's room in the first place...
"The doctor gave me a pill and I grew a new kidney!" The giddiness of the old woman and the utter bafflement of the doctors makes the scene.
Guard: How's your patient? Kirk: I think he's going to pull through. Guard: He? You went in with a she! Kirk: One little mistake...
The scene with the whalers, after they shoot. The harpoon goes 'DONG!' off the invisible Bird-of-Prey's hull, the whalers go WTF?, suddenly this huge massive ship decloaks right in front of them, and they absolutely freak out. The simple shot of the Bird-of-Prey hovering over the fleeing whaler ship is what sells it.
Everyone in the water laughing as Spock flails about...
When played in Moscow, McCoy's line "The bureaucratic mentality is the only constant in the universe." got the whole audience laughing right up to the end credits.
Shatner: On behalf of all of us, Mister President, I have been authorized to plead not guilty. On the other hand, they have also authorized me to plead guilty. Now I am confused.
Nimoy: That's good, keep rolling. Keep rolling. Ask him the question again, I like that. That was a good answer Bill.
The improv where Nimoy and Shatner stretch out the "What does it mean, exact change?" bit after a blown take is also hilarious. And there was that random scream, too.
"Ready to engage computer, Admiral. *cough*" "...Ready to engage voice?"
"Admiral, I am receiving hail storms!"
"Angels of ministers and mercy...God helps us all...to get this scene."
"Get your hand off my leg."
The Humpback Nails.
"Well, nobody's perfect." "...Oh yeah?"
Early on in the sequence at the hospital, McCoy comes across a woman in a bed receiving kidney dialysis. He remarks on the 'dark ages' of medicine he's in, then gives her a pill to take. Later, during their escape, a group of the hospital staff are standing around the woman, now in a wheelchair and looking quite lively, utterly baffled at her recovery, while she animatedly motions to everyone she can see.
Woman: The doctor gave me a pill and I grew a new kidney!