Mr. Sulu doesn't have a lot to do in this movie, but this is a sweet little moment. It's an acknowledgement of the bond between him and his two superior officers, one of those bits of dialogue that the first film forgot to include, but which hold the heart and the soul of Star Trek.
SPOCK: You are my commanding officer. You are also my friend. I have been and always shall be yours.
That gets us right in the bro-feels, Spock. This is not a guy who lies, and he wants Kirk to know that, whatever happens, he's got his buddy's back.
SCOTT: I had me a wee bout, sir, but Doctor McCoy pulled me through.
KIRK: Oh? A wee bout of what?
McCOY: Shore leave, Admiral.
This is a remarkably serious film, with blood-crazed vengeance and doomsday devices and space eels squicking their way right into our ear-canal-based nightmares. And here, in the middle of it all, is a joke about venereal diseases: a little nudge in the ribs for our rotund chief engineer.
KIRK: Well... shall we start with the Engine Room?
SCOTT: We'll see you there, sir, and everything is in order.
KIRK: That'll be a pleasant surprise, Mister Scott.
Again with the ribbing. Scotty and Kirk have been through a lot together, and even though Kirk is his superior officer, he's going to razz his buddy a little bit about the less-than-perfect way he runs his little corner of the Enterprise.
SAAVIK: (in Vulcan) He's never what I expect, sir.
SPOCK: (in Vulcan) What surprises you, Lieutenant?
SAAVIK: (in Vulcan) He's so...human.
SPOCK: (in Vulcan) Nobody's perfect, Saavik.
Spock's humor is certainly on the dry side, but this is the same kind of self-deprecating wit we see the crew of the Enterprise toss out at each other all the time. It also serves an important narrative purpose: by bringing Saavik into the fold, it makes sure we understand that she's a part of the crew now too.
McCOY: Go? Where are we going?
KIRK: Where they went.
McCOY: Suppose they went nowhere.
KIRK: Then this'll be your big chance to get away from it all.
McCoy was always the crew's straight man, with his outbursts of temper and general fussiness. The camaraderie, even in the face of mortal peril, helps make Kirk a little stronger and keep the crew tight in the face of a very serious threat.
SAAVIK: You lied.
SPOCK: I exaggerated.
Little joke, Spock? Again, we're reminded that Saavik is now a member of this merry little band, even though she's much younger.
CHEKOV: Could you use another hand, Admiral?
KIRK: Man the weapons console, Mister Chekov.
Here's a much more serious demonstration of friendship. Chekov's betrayal clearly looms large in the man's mind, even though he was compelled by the space eel in his ear and really didn't have a choice. He wants to make amends to the people he betrayed…and Kirk, mensch that he is, accepts his friend's help without a second thought.
McCOY: Are you out of your Vulcan mind? No human can tolerate the radiation that's in there!
SPOCK: As you are so fond of observing, Doctor, I'm not human.
McCOY: You're not going in there!
Spock and McCoy fight like cats and dogs, and McCoy was awfully casual with the whole "green-blooded, inhuman" thing just a short while ago. This moment confirms that McCoy never lets his temper get in the way of the deep and abiding friendship he has with this man.
KIRK: Of my friend, I can only say this. Of all the souls I have encountered in my travels, his was the most...human.
This is a riff on Saavik's earlier comment on how human Kirk is. It affirms the connection between the two men, their bond, and the fact that a man as "human" as Kirk considered the alien Spock more "human" than anyone he's ever met.