Dave Bowman's visitation to his ex-wife and his mother.
HAL thanking Chandra for telling the truth, followed by HAL, just before death, essentially asking Chandra if there is an afterlife, followed by the touching exchange between HAL and Dave:
HAL-9000: What is going to happen?
Dave: Something wonderful.
HAL-9000: I'm afraid.
Dave: Don't be. We'll be together.
HAL-9000: Where will we be?
Dave: Where I am now.
The destruction of the Discovery is a tear jerker in it's own right, doubly so as in the film version it appears that HAL has been destroyed. The novel makes it clear though that Dave saves HAL before Discovery is vaporized.
In the movie, the death of Max Brailovsky. What sells it is Curnow's breakdown:
Curnow: "You stupid bastard, get out of there!"
The movie (along with the novel) creates a retroactive tear jerker for 2001: HAL's breakdown which lead to the death of almost the entire crew of Discovery wasn't his fault. Chandra explains that the White House ordered HAL to keep the existence of the Monolith a secret, which conflicted with his basic programming. This caused him to descend into paranoia which resulted in him attempting to kill the crew. Even worse is that Floyd, the man who discovered the Monolith in 2001 and planned the Discovery mission, had no idea that this occurred. He's understandably livid when Chandra reveals the truth.
Chandra: HAL was told to lie... by people who find it easy to lie. HAL doesn't know how, so he couldn't function. He became paranoid.
Floyd: Those sons of bitches. I didn't know. I didn't know!
The recorded video message about the "state of war" now existing between the USSR and the USA. Milson, the head of the NCA, and his Russian counterpart ordering the American and Soviet teams apart from each other, complete with the separating of the bridge between them, is very sobering, after seeing the astronauts from both nations collaborating and becoming friends.