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.
Kirk brings up the fact that the inhabitants of 20th century Earth will have never seen an extra-terrestrial before. Cue pointed looks at Spock's eyebrows and pointed ears. Spock, never changing expression, tears off a strip of his robe and wraps it around his head to hide said eyebrows and ears, with an 'Any further objections?' air. Bones' bemused smile and headshake seals it.
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 does a 'Hell if I know' gesture.
Kirk's extended reaction when he sees Spock in the aquarium tank - mind-melding with the whales - is priceless. First he glances at where he thinks Spock is, repeatedly looks around the tour group wondering what the hell happened to him, his Jaw Drop when he sees Spock in the tank itself (clearly thinking "Spock, are you out of your Vulcan mind!?" at some point, probably all of it), to his 'Oh, just what we needed' look at the tour woman who finally points this out to Gillian.
Any time Spock tries to use "colorful metaphors" (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.
Doubly funny since that actress was an extra who signed up because her car got towed (she didn't know they were making a movie when she parked it) and she figured her day was shot anyways, so she might as well make something of it. She didn't know that she wasn't supposed to interact with anybody. Watch their faces when she does. The scene was so brilliant that it was left in.
With that motorcycle cop just glaring at Chekov the whole time he's asking everybody where the nuclear wessels are. That was a real cop with no idea he was in a movie, by the way.
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...
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!
If you look closely, you can see her waving to McCoy and he smiles back.
Shortly after when they are knocking over patients and doctors, including one with a broken leg, Bones stops to help them too, only for Kirk to pull him back.
Easy to miss joke, but the hospital PA pages a "Dr. Ben Dover." Say it out-loud.
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?"
At Plexicorp Scotty has asked if his "assistant" (McCoy) can join him on the tour. He assumes an amused smug expression so Bones tries to deflate him by saying "Don't lose yourself in the part!" Scotty's smile doesn't waver one iota.
After arriving, Kirk realizes that they need money. He takes his antique glasses (Bones' gift from Star Trek II) to a dealer, and is told that with the lenses broken, he can get $100. Cue Kirk's reaction: "Issssssss that a lot?" The dealer's response is a shrug and a smile.
The answer to "What was the principal event of the year 1987?" is apparently "Computers are cloned from carrots on Earth." (The film was released in 1986.)
"Gracie is pregnant." What sells it is how casual, matter-of-fact, and out-of-the-blue Spock is when he says it, followed by Gillian stomping on the brakes.
"The rest of you stay here." Kirk then turns around and sees everyone else just standing there, and amends it to "The rest of you break up. You look like a cadet review." The others...just kinda move around a bit, prompting a "Just forget it" reaction from Kirk.