Episode 23. Wolfwood's speech gives this troper the sniffles.
The end of the post-credits scene in the final episode, when Meryl sees Vash come back.
In the manga, the part in Vash and Knive's past where they discovered the horrors of the previous experiments and, after many days of refusing to eat or touch anything from humans, Rem manages to get through to Vash. Especially how Vash (shown surprisingly to be even more rash and violent than Knives in his youth) tried to kill himself with a knife only for Rem to grab it and get stabbed, with him muttering how he won't feel sad about her death and how he won't cry... only to break down sobbing and holding her.
The preview for episode 18, which is both heartwarming and tearjerking at the same time:
Kid!Vash: Rem said she came to this ship because someone she loved back on Earth died. I tried to imagine what would happen if Rem died. Suddenly, hot water began to flow from my eyes and didn't stop for a long time. Rem, don't ever leave.
In the manga, Vash's exchange with Meryl before he heads off to fight Legato and Knives.
Vash and Knives flying away while basically hugging at the end of the manga (while it also has creepy overtones). Knives's epiphanic revelation that he's "the one who grew apart."
Episode 25, when Meryl stops the townspeople from torturing Vash. She repeats Rem's line: "No one has the right to take the life of the others." What makes it heartwarming is that Vash sees her as Rem when she speaks, because he has been holding that ideal in his life.