The way Hogarth whispers "I love you," as The Giant takes off.
"Souls don't die..."
The Giant after believing Hogarth to be dead. Even savvy viewers that knew otherwise got choked up by the Giant grieving for his friend. Especially when he starts trembling and buries his face in his hands.
If you listen close during that scene, you can hear the Giant whimper, "Hogarth."
Watching the Giant succumb to his defense mechanism can be a Tear Jerker as well. Yes it can be a Awesome Moment for some, but it's very much like Sulley scaring Boo; we've gotten so used to this Gentle Giant that seeing him like this is legitimately horrifying, especially since he actually tried to resist his mechanism just a few seconds ago.
The deer in the forest and Hogarth's explanation of the concept of death to the Giant.
Hogarth: Things die...It's a part of life. It's bad to kill. But it's not bad to die.
This exchange of dialogue:
Annie: Shouldn't we get to a shelter?
Dean: ...it wouldn't matter...
Just the hopelessness and sense of realism in his voice...
The scene after Hogarth nearly gets vaporized by the Giant when they were playing a game and the Giant mistook Hogarth's toy gun for a real weapon, and Dean yells at him, calling him "a big gun that walks." Giant then gets a "My God, What Did I Almost Do?" moment when he sees a bus with a vaporized hole as Dean tells him, "You almost did THAT to Hogarth!"