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Tear Jerker: Anastasia
Anastasia's dance with the Czar, her father, during her Disney Acid Sequence dance. No matter what opinion you have about Czar Nicholas II, the flashbacks and visions involving the family, Alexandra (the mother), the older sisters (Tatiana, Olga and Maria), and the little brother (Alexei), strikes as heartwrenching considering what happened to the family in reality.
The reunion between Anastasia and her grandmother Marie.
Dimitri's reaction when he realizes Anya's true identity. The girl he's fallen in love with and thought was his equal is the real Grand Duchess. And they can't be together because he's a former servant and is now unworthy of her.
Perfectly shown in his defeated conversation with Vlad.
Vlad: That means Anya has found her family! And you—
Dimitri: Will walk out of her life forever.
Dimitri: Princesses don't marry kitchen boys.
Later scenes add to this: Him being forced to bow to her, refusing the money and his expression when he sees Pooka holding her crown. No one's telling Dimitri he's not good enough for Anya - he just knows he is. The instinctive values here can't be emphasized enough. He was a servant at the beginning who admired the Princess but was never noticed. He obviously spent years trying to move past that, and now, because of a difference in title, Anya is unreachable again. They're still the same people, she's still the same girl he's insulted and saved and comforted but suddenly that's it, he's just as inferior to her, as when they were children. And he can't do anything about it.