The first ten minutes. Yes, ALL of them. If you know anything about the film at all, you know Kirk the elder is going to die. Giacchino's music is what really sells it.
The expression on his face when George Kirk realizes that the Kelvin will have to be manually piloted and rammed into the Narada. I'm not sure if there's a section for the "Moment of Realization of Self-Sacrifice," but there should be!
Chekov, when he loses the transporter lock on Amanda. It's Anton Yelchin's heartbreaking facial expression that makes it.
It's worse when you realise that he himself had recognised his Crowning Moment of Awesome from earlier, and so had assumed he'd perform the miracle twice. Failing to do so - and so shortly afterwards - must've absolutely crushed him.
Also taking into account the way he shouts "I'm losing her! I'm losing her!" and then, once he has, he starts whispering "I lost her... I lost her..." in a voice so quiet and broken that you almost can't hear him.
Kids, he's only 17 years old and fresh out of the academy. This whole sequence was one big wake-up call that probably kept him awake at night for a while.
That heartwrenching scene when Spock is trying to appear stoic after the destruction of the planet Vulcan. Uhura keeps saying, "I'm so sorry" over and over again. She offers to hug Spock- surprisingly, he accepts it, and for a heartbreaking moment, he buries his face into Uhura's shoulder, looking like he's going to cry. Watching a crying Uhura comforting a devastated Spock was the saddest part in the film.
"I am now a member of an endangered species."
Kirk: "So I have to somehow make Spock emotionally compromised." Old Spock: "Jim, I just lost my planet. Believe me, I am emotionally compromised."
When we finally see Spock emotionally compromised, it is probably the saddest part of the movie to see the stoicism break to the point that he nearly KILLS Kirk. For a second, it seemed not even his father could stop him...
The entire character of Spock Prime. Think about it. Spock Prime lived through the entirety of TOS and the movies that followed it. Based on his age, it's been years since Kirk's death. Who knows how many others of the original crew have died. He loses Romulus, all because he was too slow to get there, and is burdened with the guilt of that. Then, he goes through the rip and into the alternate universe. He's gone back in time, but everything's different. He loses his entire planet, and you can see that as being his fault as well. And, on top of that, he finally sees Kirk again, and everyone else on the crew, but he knows that they're different people, and that he can't interfere because he wants the alternate Spock to live out those adventures. It all boils down to him being laden with guilt for the loss of two planets, and very, very alone.
It gets worse when you remember the TNG 2-part episode "Unification", in which Spock was working towards peaceful relations between Vulcans and Romulans, and the destruction of Romulus ruined all of it in more ways than he could ever imagine.
A lot of people gave Kirk's death scene from Into Darkness flak, saying "Oh, it's just Wrath of Khan's but with the roles reversed! They even use the same lines!" And, in a lot of ways, they're right. However, one crucial difference between Into Darkness and Wrath of Khan is that Spock handled his death stoically and, while clearly saddened that he was about to die, he was able to reassure Kirk that he was at peace and accepting of his demise. Kirk, on the other hand, demonstrates that he does know that he did a great thing in saving the Enterprise but, at the same time, he's so goddamned scared of dying that he's clearly panicking as his radiation poisoning overcomes him. And instead of Spock comforting Kirk about his own death, Spock is instead struggling to hold back tears and, at the same time, comfort Kirk as his best friend slowly dies in front of him. So yeah, while the two scenes are incredibly similar, they both have a crucial difference that shows just how things would have been had the roles been reversed in Wrath of Khan.