Kirk and McCoy have convinced one of the bad guys to finally explain what the hell's been going on... just in time for Enterprise to find them and beam them up. Once Kirk realizes what's going on, he shouts "Son of a bitch!" and disappears on the last word, leaving it to echo for a while longer.
Chekhov: You want to go back?
McCoy's reaction to General Chang's incessant hamming is gold:
I'd pay real money if he'd shut up.
"If I were Human, I believe my response would be... *Fascinating Eyebrow* 'Go to Hell.' If I were Human."
When Kirk tackles the Federation President at the end of the movie (to save his life, of course) he briefly introduces himself as "Kirk, Enterprise" while he's basically lying on top of the man. It's also the super-casual way he says it that makes it hilarious. One wonders if he meets his blind dates that way.
Speaking of blind...the Federation President was supposed to be blind (the glasses he puts on to see the Operation Retrieval papers was a prototype of the VISOR Geordi wears on TNG), so after flying-tackling a blind man as weapons are being fired, Kirk thinks this is the way to help the man calm down!
McCoy is on the witness stand in Klingon court and is asked his current medical status. He makes a joke about how it's fine, aside from a touch of arthritis. A SINGLE KLINGON laughs his head off while the rest of the assembled audience is quiet. Yes, some random Klingon in the peanut gallery thought that was the funniest thing he's heard all day.
The ship is scoured for the missing gravity boots and finally found among the belongings of a crewman. Chekov smugly deposits the boot in front of the crewman, with the declaration "If shoe fits, wear it." Problem is, the shoe DOESN'T fit, as the crewman can't wear standard issue boots because of the unusual shape of his feet.
Rather a cruel joke, but in the last time Uhura ever appears in the credits of any Star Trek production (barring the 2009 movie), it's misspelled.
Spock convinces Scotty to waste time in order to delay the Enterprise's return to Spacedock.
Spock: Ah, Mr. Scott, I understand you're having trouble with the warp drive. How much time do you require for the repair? Scotty: There's nothing wrong with the bloody thing...! Spock: (looming over Scotty to press the urgency) Mr. Scott, if we return to Spacedock, the assassins will surely find a way to dispose of their incriminating footwear and we will never see the captain or Dr. McCoy alive again. Scotty: ...could take weeks, sir.
After Chekhov wonders why the conspirators couldn't just "waporize" their uniforms and magnetic boots, Valeris demonstrates why by phasering a cooking pot out of existence, setting off an alarm. The Oh Crap of the galley crew is funny enough as it is, but then Scotty and Uhura both come in through the end of the scene asking if someone fired off a phaser aboard the ship.
Uhura: Did somebody fire off a phaser?!
Scotty:(angrily) Who fired a-?!
A bit of Metahumor and Stereotype Flip during the briefing scene early in the movie. When Admiral Cartwright, one of the few black characters in the movie, predicts that the Klingons will become the "alien trash of the galaxy", watch Southern Gentleman Doctor McCoy's "dafuq?" reaction. Given The Reveal about Cartwright's part in the plot, the humor wears off later in the film.