Thomas Harewood commits a suicide bombing for Khan in exchange for his daughter being cured of her illness.
Pike's death. It's not even a typical death scene. Pike doesn't give any parting words; no speech, no nothing. He simply stares uncomprehendingly at Spock, clearly scared out of his mind. And then Spock does a mind meld with him, and Spock's face morphs just enough that we get an idea of what Pike is feeling.
You could tell Bones is well aware of just how hard Pike's death affected Jim.
The look on Scotty's face when Kirk accepts his resignation. He's just been utterly dismissed by his captain, and lost his ship and his career, all for trying to do the right thing.
Spock and Uhura's argument about his apparently not caring if he died in the volcano in the opening scene. Spock is eventually driven to admit that he forced himself not to feel anything when it seemed he was doomed, as he never wants to feel even a fraction of his emotional reaction to the destruction of Vulcan again.
It's brief, but look at the expression on Spock's face when Uhura offers to confront the Klingons herself.
Kirk, who has spent all of both movies having every one of his snap decisions, wild guesses, and gut feelings turn out for the best, turns and looks at his crew when he realizes that Admiral Marcus isn't going to let any of them live no matter what he says or does. The hopelessness and defeat in his voice is so jarring.
Also making it worse is Uhura's gasp at Kirk's defeated apology, showing just how hopeless the situation is.
Kirk decides he needs Harrison's help in order to defeat Marcus, so goes to talk to him. Spock follows, trying to talk Kirk out of it. Then Kirk admits to what Pike was trying to tell him after the Nibiru mission; that he's not ready to be a captain.
Spock: I cannot allow you to do this! It is my function aboard the ship to advise you in making the wisest decisions possible, something I firmly believe you are incapable of doing in this moment.
Kirk: You're right! What I am about to do, it doesn't make sense, it's not logical; it's a gut feeling! I have no idea what I'm supposed to do. I only know what I can do. The Enterprise and her crew needs someone on that chair who knows what he's doing. That's not me. It's you, Spock.
There's something to be said for Spock asking his time-displaced older self about his encounters with Khan in the prime timeline. You can tell even a hundred years later, Spock Prime is still haunted by what it took to defeat him the first time.
This scene is much Harsher in Hindsight because not only was this Leonard Nimoy's last scene as Spock in a Star Trek movie; this was his last movie role ever. This means that Nimoy's last movie scene of his lifetime was him as his most iconic character warning his younger self of the hardships he's faced over the years and remembering his own famous Heroic Sacrifice. Truly a perfect Passing the Torch moment and a fitting epitaph to one of the greatest actors of our time.
The moment where Carol is pleading to her father to spare the Enterprise as she's still on it by forcing him into a Sadistic Choice, only for him to sidestep it and teleport her aboard his own ship, the Vengeance. The fact that she still pleads even when it's apparent she can't affect anything is just depressing.
And she gets her leg broken by Khan and watches her father die right before her eyes. Poor woman needs a hug.
Kirk going to his death to repair the damaged reactor room while Scotty watches on.
There's also Spock's reaction when Scotty calls him down to the warp core room. He doesn't even know what it is, but he knows something terrible has happened to Kirk and he legs it so fast, he doesn't even give the legally required order that someone take over. Then you see him running through the halls, and the sheer, utter panic on his face is something to behold.
Spock desperately pleading to Scotty to let Kirk out, but Scotty sadly says he can't do it because it would kill everybody in Engineering. Then the two have a final conversation, and Kirk dies.
Kirk admitting to being scared and asking Spock to help him not be. Spock, who is crying at this point, tells him that he doesn't know how to anymore.
Kirk:I'm scared, Spock. Help me not to be. How do you choose not to feel?
Spock:(in tears) I do not know. Right now, I am failing.
Kirk:I want to tell you why I couldn't let you die. Why I went back for you.
Spock: (tear falls from his cheek) Because you are my friend.
Kirk nods and puts his hand on the glass, and Spock does the same, mirroring the famous Wrath of Khan pose, and Kirk dies.
It's the famous pose which really drives it home. Those who had been able to hold back their tears until then just lost it at this point.
Followed by Uhura showing up just in time to see Kirk take his last breaths. She remains teary-eyed until Spock beams down to catch up to Khan.
After Kirk dies, you can see Spock's grief turn into pure rage.
The track for this scene, "Buying the Space Farm." Just...listen to it. It's Spock's emotions put to music.
Not just put together. It's not just that the scores are over-layed together; they weave in and out with one character's motif taking the lead before it fades back while the other character's comes to the forefront, before finally combining into a single whole.
Spock crying. When the Vulcan loses it, we don't stand a chance.
When Kirk dies, Spock has this very sharp intake of breath, like he still can't believe this just happened.
Spock's pursuit of Khan. Yes, it's awesome and incredibly badass, but it's a bullet to the heart. We got a glimpse of this level of rage in the first movie, but seeing Spock lose control so thoroughly and for so long is hard to handle. He even attacks Khan with a mind-meld, using either his physical pain or more likely his own rage and grief to incapacitate his opponent. How many Vulcan laws did he break with that move?
McCoy seeing Kirk's body and just walking away, sitting at his desk, and breaking down. The look on his face is just so utterly heartbreaking.
Made even worse when you remember that McCoy was Kirk's first friend at the Academy, and they were shown to routinely hang out while there. They've spent years together and he's just seen his friend's body, and it's too late to save him or so he thinks.
Worse still when you consider the amount of times McCoy tells Kirk to be careful or expresses worry for him throughout the movie.
Harrison: Marcus took my crew from me! [...] He used my friends to control me. I tried to smuggle them to safety by concealing them in the very weapons I have designed. But I was discovered. I had no choice but to escape alone. And when I did, I had every reason to suspect that Marcus had killed every single one of the people I hold most dear. So I responded in kind. My crew is my family, Kirk. Is there anything you would not do for your family?
His anguish when he thinks he's killed them all, too.
There's a small moment, when Harrison is confronting his ally Admiral Marcus, and he's just seething with rage at the betrayal and manipulation. But what does he say? "I will destroy everything you've ever loved?" "I will conquer the stars with your own ship?" No. "You should've let me sleep." Deep down, Harrison knows who and what he is, and how his time has passed, and trying to use him in this new era is a mistake. While he wants to survive, he knows it would be safer, for everyone, if he was just left alone.
Although, considering Harrison's rage during that scene, one could easily have taken the unstated above as implied, which pushes it from Tear Jerker to Nightmare Fuel.
Take a note of Harrison's last scene. Harrison is smiling. He may be a horrific monster, but he is back with his family, and at peace.
No matter how you look at it, this film gets the really sad parts done right. There is nothing said, the music is only there to help create a mood, and then it gets soul-crushing when it vanishes. This film follows "show, don't tell" for the truly sad parts.