The film has perhaps the only one that happens before the film even starts: it was released soon after the Challenger disaster, and the producers took it upon themselves to put the following at the beginning of the print: "Dedicated to the crew of the space shuttle Challenger, whose courageous spirit will live to the 23rd century and beyond."
The conversation between Spock and his mother Amanda is wonderful:
Amanda: Spock, would you say that the needs of the many, outweigh the needs of the few?
Spock: Of course. I would consider such a statement an axiom.
Amanda: Then you stand here, alive, because of a mistake. Made by your flawed, human friends who believed the needs of the one—you—was more important to them.
Spock: . . .humans make illogical decisions.
Amanda: [fond smile] They do indeed.
When Chekov informs Kirk he and Uhura have found the "nuclear wessel", Chekov notes "Admiral, it is the Enterprise.", which almost takes Kirk aback, that no matter what time or place they're in, the Enterprise will be there to help him.
The scene in the hospital where McCoy gives the sick old lady on dialysis some 23rd century medicine... and a few scenes later, she's grown a new set of kidneys. If you look closely, you can see the woman waving to McCoy and he smiles back.
Kirk slipping into First-Name Basis with Chekov while rushing him out of the hospital.
"[pushing his head back down] Not now, Pavel."
This exchange between our main trio:
Spock: Mister Scott cannot give me exact figures, Admiral. ...So I will make a guess.
Kirk: A guess? You, Spock? That's extraordinary!
Spock: I don't think he understands.
McCoy: No, Spock. He means that he feels safer about your 'guesses' than most other people's facts.
Spock: Then you're saying ...it is a compliment.
McCoy: It is.
Spock: Ah. Then I will try to make the best guess I can.
This is a call back to an earlier scene where Spock is troubled about not having exact figures, and Bones suggests for him to guess. When Spock flatly says that it's not in his nature to guess, Bones smiles and tells Spock that "Nobody's Perfect" which Spock apparently takes to heart.
The scene where Kirk and the others talk about rescuing Chekov. They ask Spock what he thinks, and he agrees that they must save Chekov. Kirk asks, "Is that the logical thing to do, Spock?" Spock answers, "No, but it is the human thing to do." At that moment, Kirk and everyone else knew that Spock was truly back.
"Admiral! There be whales here!"
The waterfight at the end. Particularly when Kirk picks Spock up and throws him bodily into the water.
Keep your eye on Spock in the scenes leading up to that. Is he... smiling? He might be, but he'll never tell!
The scene where the Enterprise crew are put on trial:
President: Captain Spock. You do not stand accused.
Spock: Mr. President, I stand with my shipmates.
Sarek HATED the idea of Spock joining Starfleet. But after all that, he pulls him aside at the end and admits he was in error.
Really, the entire conversation is heartwarming in at least three different ways. Sarek shows his love for Spock, both unconditionally and with direct approval for his life choices; Kirk and the crew earn the respect of the reserved Vulcan (validating the entire intercultural-cooperation idea of the show, really); and Spock acknowledges his deep friendship with them. All with the typical reserve and formality of Vulcans, yet immensely powerful thanks to the performances of Mark Lenard and Leonard Nimoy. Video!
Sarek: I am returning to Vulcan within the hour, I would like to take my leave of you.
Spock: It was most kind of you to make this effort.
Sarek: It was no effort. You are my son. Besides, I am most impressed with your performance in this...crisis.
Spock: Most kind.
Sarek: As I recall, I opposed your enlistment in Starfleet. It is possible that judgement was incorrect. Your associates are people of good character.
Spock: [matter-of-factly] They are my friends.
Sarek: Yes, of course.
It's capped with Spock finally asserting his humanity and closing the bookend from the beginning of the movie, when he tells his father to tell his mother "he feels fine".
Sarek's confidence that the Enterprise crew will pull through and fix the problem is pretty heartwarming, and further shows how much he's come to respect them as well as his son.
Whenever he looks out the windows, it's not so much "How much worse will the storm get?" as it is "Kirk, quit screwing around and save the day already!"
For the Trekkie audience, seeing Jane Wyatt make her appearance as Spock's Mom was heartwarming itself.
And of course Trek's greatest Heartwarming Moment ever — Kirk and company think they're being reassigned to the Excelsior, but just as Alexander Courage's classic fanfare plays we see...the brand new USS Enterprise, NCC-1701-A! Along with the classic quote, "My friends, we've come home." Trek fans weep and cheer every time at that scene.
Especially pay attention to Scotty, as Uhura a hand on his shoulder and he looks almost on the verge of tears.
But then there's the outtake of that scene, wherein there is a moment of silence and then Leonard Nimoy deadpans, "Get your hand off my leg."
Heartwarming in Hindsight: This movie helped truly kickstart cetacean protection programs. As a result, the humpback whale, the species featured in the film, has seen their numbers octuple since the movie was released.