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?"
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.
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.
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.
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."