After Spock compares V'Ger to a child, McCoy replies "This 'child' is about to wipe out every living thing on Earth. Now, what do you suggest we do, spank it?"
A few moments later, when V'Ger is attacking the ship "Your 'child' is having a tantrum, Mr Spock".
McCoy again, after Spock rejoins the crew:
Spock: On Vulcan, I began sensing a consciousness more powerful than I have ever encountered — thought patterns of exactingly perfect order. I believe they emanate from the intruder. I believe it may hold my answers. McCoy: Well, isn't it lucky for you that we're heading your way? [does hitchhiking gesture] Kirk: Bones!
Kirk: Well, for a man who swore he'd never return to Starfleet— Bones: Just a moment, Captain, sir. I'll explain what happened. Your revered Admiral Nogura invoked a little-known, seldom-used reserve activation clause. In simpler language, Captain, they drafted me!
For those with a military background, it's appropriately amusing to know that even two hundred and fifty years in the future, Stop-Loss policies and orders still wreak their havoc on poor servicemembers' retirements.
Outtake first seen in the ABC-TV version:
After the transporter is fixed, an arriving crewman says "Someone wanted to first see 'how it scrambled OUR molecules'".
Kirk to Transporter Chief Rand: That has a familiar ring to it....note Yep. Bones again.
Given that this happened only a few scenes after Sonak's demise in that very same transporter, it's amazing that Kirk or the audience can find that funny.
Uhura explaining (while having a hard time keeping a straight face) that "one" of the extended crew members is refusing to beam aboard and Kirk (with a barely contained straight face himself) saying he will make sure that crew member beams aboard. They both know damn well who it is.
Not intentionally funny, but Bones appearing on the ship in a funky 70s robe-like uniform, full beard and "the eye piece ofthe Staff of Ra" around his neck definitely elicits a laugh.
His clothing and demeanor combine to give the impression not that Bones was recalled to Starfleet and was standing by to come aboard, but was instead in a retirement home and got beamed up with no warning on the way to the bathroom.
In a rare aversion of his usual tendency to react in entirely the wrong way to a potentially dangerous situation and get injured as a result, Chekov's reaction when he's ordered not to interfere with the probe—which is clearly pursuing him while he tries (unsuccessfully) to retreat from it—is very funny. It's the look on his face that really sells it, but the way the line is delivered is hilarious too.
Chekov:Absolutely I will not interfere with it!
Even funnier when you consider that in the original script this was Chekov's death scene. The fact that his whole demeanor is clearly asking Who Would Be Stupid Enough? to try to touch the probe (or, more accurately, why everyone else assumes he will be stupid enough) makes the inversion of his usual naive behaviour even better.
In a scene cut from the original but restored for the expanded TV and Director's Cut versions, soon after the above exchange, Sulu becomes flustered in Ilia's presence, even pushing the wrong button on his console. Takes on an additional tone of irony in that the actor is gay.
Kirk and Decker discuss the possibility of manipulating the Ilia-probe's imported memories of Ilia's life, friendships, and loyalties, in order to complete their mission...
Kirk: Commander! Will, we're locked in an alien vessel, six hours from Earth orbit. Our only contact with our captor is that probe. If we could control it, persuade it, use it...
After escaping the wormhole, Chekov reports no casualties, to which McCoy retorts:
McCoy: Wrong, Mister Chekov, there are casualties. My wits! As in "frightened-out-of", Captain, sir!
A little moment, but when Kirk tries to interfere with Decker directly interfacing with V'Ger, the Ilia!Probe nonchalantly pushes him away, making him reel ten feet, as if to say, "Back OFF! That's my MAN."
A moment that unfortunately did not make it into the film, but is detailed in Star Trek Movie Memories; Leonard Nimoy did not like Spock's simple declaration that he didn't need to be returned to Vulcan, having already finished his business, so he ad-libbed a line: "Negatory, Captain. If Dr. McCoy is to remain aboard, my presence here will be essential."