Tarzan's parents and what happened to them. Picture it: Victorian-age Englishfolk, lost in the rainforest with their infant son after having miraculously survived a shipwreck. Yet they're determined not to just lie there and die. They built a treehouse, working together (when the beam slips on the father's shoulders and the mother joins him to help carry it, that's when the tears start). It's a pretty impressive-looking treehouse, and it seems as though the three of them will at least be able to live comfortably... and then Sabor shows up.
The death of Kerchak, right after he finally accepts Tarzan as his son.
While the focus for most of the scene is on getting Jane, the Professor, and Clayton away from him, Kerchak's reaction to seeing that Tarzan has defied his orders and brought them to the home he was trying to protect is saddening - especially since just before they showed up, the two had just had the beginnings of an understanding. When he first sees them, he just seems too stunned and confused to react, and only flies into a rage once Clayton pulls out his gun. He doesn't even look angry at Tarzan afterwards, just betrayed, his voice full of hurt.
Kerchak: I asked you to protect this family. And you...betrayed us all.
After Kala takes Tarzan back to the treehouse she found him in, and he finally knows the full truth of his past:
Kala: Tarzan... I just want you to be happy. Whatever you decide... Tarzan: No matter where I go, you will always be my mother. Kala: And you will always be in my heart.
The fact that Tarzan is dressed in his father's clothes during that scene doubles the emotional impact. Something about a well-dressed gentleman hugging a gorilla and reassuring her that she is his mother just increases the poignancy.
Phil Collins' "You'll Be In My Heart" instrumentally played in the score during this scene reeeeaaaally gets the eyes to leak.
Kerchak scolding Tarzan for causing the elephant stampede.
When Jane and Tarzan say goodbye at the end of the film. Yes, it only lasts for a couple of minutes, but the heartwrenching tension is PAINFUL. And then Disney thinks, "Oh wait, this scene isn't emotional enough," so what do they do? Make Jane lose her glove in the wind and conveniently fly over to Tarzan, who picks it up and gives Jane one of the most pitiful looks in the history of Disney animated canon. Oh GOD.
"You came back..." "I came home."
The moment when Tantor and Terk think they've missed saying goodbye to Tarzan. Terk flies into a sorrowful rage while Tantor simply says, "I'm gonna miss him too."
Tarzan has spent his entire life not knowing why he's so different from his mom, his dad, and his friends. He has no idea who or what he is, and then all of a sudden three humans show up...and he finds out Kala and several other gorillas had known about humans all along. The anger in Tarzan's voice when he confronts Kala about the humans is just heartwrenching.
Tarzan: Why didn't you tell me there were creatures who looked like me?!
He also takes Clayton's betrayal especially badly, which makes a lot of sense. While it's not as focused on as his relationship with Jane, it's clear that as the only people even remotely like him he's ever met he's latched on particularly hard to Professor Porter and Clayton as well, and trusts Clayton enough to take his advice without question. The look on his face when Clayton reveals himself to be with the mutineers cycles from fear, to confusion, to pain and rage in a very heartbreaking way.
Note in spite of all this Tarzan didn't have it in him to kill Clayton, and when the latter attempts to slash through the vines he's entangled him in, Tarzan outright begs him to stop before it's too late. He looks pretty traumatised by the sight of Clayton's corpse dangling in front of him, having accidentally set up his own demise.
That heartbreaking scene where a young Tarzan tries to cover himself in mud to be a gorilla...god...he just wanted to fit in.
The Musical adds in a reprise of "You'll Be in My Heart" (sung as a duet between Kala and Tarzan) after Tarzan decides to leave the jungle and return to society to be with Jane. While the musical was seen as So Okay, It's Average, every critic admitted that that scene would reduce the entire house to tears every time.