This is based on opinion. Please don't list it on a work's trope example list.
Tear Jerker / Rise of the Planet of the Apes
Scenes with Will's father that reveal the symptoms of his Alzheimer's disease, especially after Cure 1.0 fails.
The death of Will's father, who refuses to take Cure 2.0, signalling to his son that it was perhaps time to let go.
Arguably made worse by the fact that the scene is completely wordless.
Caesar being separated from Will for the first time at the primate shelter.
Later, Will returns to get Caesar, and Caesar acts like he's angry at him and refuses to go with him. In reality, he'd bonded with the other apes in the shelter and had a mass escape plan prepared, and he could not abandon them. When Will leaves, we see Caesar's expression showing his true feelings and just how tempting the prospect of going back with Will really was for him.
Caesar learning that his mother died and that he is virtually a freak of nature different from humans and other apes.
Due to the Downer Ending described in the Main section, the realization that Caesar will likely soon lose his adoptive father as well.
Franklin's death. He was one of the only people at the facility who actually cared about all the apes, yet he dies a slow and probably agonizing death as the first victim of the plague.
The immediate aftermath of Caesar attacking Hunsiker to protect Charles. After Hunsiker's daughter comes out, Caesar realizes that he made a mistake, and goes over to hug Charles. The expressiveness on his face is just gut-wrenching, as seen at the top of this page.
Everything about the attic window that Caesar draws on the wall of his cell.