When Spock is resurrected at the end. The entire point of the movie comes to an emotional head with the following exchanges.
Sarek: Kirk, I thank you. What you have done... Kirk: What I have done, I had to do. Sarek: And at what cost? Your ship. Your son. Kirk: If I hadn't tried, the cost would have been my soul.
and shortly after ...
Spock: My father says that you have been my friend. You came back for me. Kirk: You would have done the same for me. Spock: Why would you do this? Kirk: Because the needs of the one... outweigh the needs of the many. Spock: I have been and ever shall be your friend. Kirk: Yes. Yes, Spock. Spock: ... Ship, out of danger? Kirk: You saved the ship. You saved us all. Don't you remember? Spock: ... Jim. Your name... is Jim. Kirk: Yes.
The fact that one of the first things that Spock spontaneously remembers about Kirk is that his name is "Jim" is a CMOH in and of itself.
At the point of no return, Kirk tries to shield his friends from the danger and other possible consequences:
Kirk: My friends...I can't ask you to go any further. Dr. McCoy and I have to do this. The rest of you do not. Chekhov: Admiral, we're losing precious time. Sulu: What course, please, Admiral? Kirk: Mr. Scott? Scotty: I'd be grateful, Admiral, if you'd give the word. Kirk: (touched and humbled) Gentlemen, may the wind be at our backs. Stations, please.
It seems that at least one of the writers of Star Trek: The Next Generation novelizations enjoyed this scene too. It's likely that a scene in the novel Ghost Ship is a take on it. When Picard needs someone to command the saucer section so the 900+ passengers of the ship can escape the destructive world-eating entity bearing down on them, (while the battle section distracts it, giving itself only an 18% chance of survival), not a single member of his bridge crew volunteers to take the mission - which would offer them a better chance of survival. Picard gets as choked up as the stoic leader ever does.
Picard: ...I'm very proud of all of you.
When Spock's soul is being put back into his body, the Vulcan High Priestess explains the dangers and small chance of success to Sarek. She then asks him, "Is this logical?" He replies, "My logic is uncertain where my son is concerned." Mark Lenard's subtle expression and intonation speak quiet Vulcan volumes.
During the ceremony, couple of quick shots show Saavik and Sarek with their heads bowed as well. Every Vulcan in that temple was pouring their very soul into the effort to bring Spock back.
And the scene between McCoy and Spock's katra-less body on the way to Vulcan, when the crusty doctor finally unbends enough to admit the depth of his friendship and grief. One of DeForest Kelley's finest moments.
Bones: I'm gonna tell you something that I... never thought I'd ever hear myself say. But it seems I've...missed you. And I don't know if I could stand to lose you again.