When Anastasia thinks that Dimitri is dead and cries over him, only to discover that he was alive, her reaction is deeply touching.
Anastasia being reunited with her grandmother.
Also, when Anastasia explains to Sophie how she escaped from the Palace and mentions that a boy who worked in the kitchen saved her from the soldiers. This shocks Dimitri — because he suddenly realizes that he is that boy, and it's THE proof (aside from a certain key) that Anya is truly Anastasia. She can barely recall her early years, but she still remembers him even when she doesn't know that the man in front of her is the same boy who rescued her.
Dimitri's determination to prove Anya is the real Princess. Even though he'll lose the woman he loves and be seen as a kitchen boy again, he still does it. He's suffering because yes, he loves her.
Dimitri comforting Anya after she wakes up from her sleepwalking nightmare.
Vlad has Anya and Dimitri practice waltzing on the ship to a romantic reprise of "Learn to Do It." The lyrics are especially pretty:
Vlad:I taught her well, I planned it all, I just forgot... romance.