Spirited Away/Heartwarming

Haku giving Chihiro the pill that stops her disappearing, and then demonstrating how it works by them gently touching hands. It's easy to miss, but Haku's expression is extremely tender and shows the first implications that he's not as cold as he makes out to be. Helped by the very sweet music.
When Chihiro tells Haku his true name, he turns back into a boy and they perform a Headbutt of Love while flying through the air. This troper once read a review that described this scene as, "they fall, literally and figuratively, into each other."
It quickly becomes evident that Lin isn't a grumpy or mean as she originally appears. She seems delighted to hear the Sen got her job, wades through swamp muck yelling that she's coming and she won't him hurt Sen, and even delivers a platonic "If You Do Anything To Hurt Her" at Noh Face.
The ending of the English dub version, where the last lines are Chihiro's cutely matter-of-fact "I think I can handle it." as regards the prospect of transferring to a new school. Even if her parents remembered and learned nothing from the ordeal, she did. This is a girl who just faced down practically the entire spirit world by power of will and virtuous behavior... real life is so going to get its ass kicked, and you know it.
At Zeniba's house, Chihiro is sick with worry for her parents and Haku. Suddenly, the windows rattle and with a smile, Zeniba tells Chihiro that their guest is on time. Chihiro goes and opens the door... to see Haku, in all his perfect, dragon-formed glory. There are no words to desrcibe the joy on Chihiro's face as she runs up to Haku and gives him a tight embrace.
And before that, Chihiro needed to confront No Face and Zeniba, so she told an unconcious Haku to not die and that she'll back soon

Lin: "What's going on?"
Kamaji: "Something you wouldn't recognize. It's called love."

Chihiro and her companions have just gotten off the train, and are standing there, probably wondering what to do next. A lamp with a foot hops squeaking out of the forest and bows to them, then hops away again. They follow it. A moment later, the camera cuts to a farther-back view, and the most beautiful, heartwarming little tune starts playing. That was the exact second that this troper, upon seeing the movie for the first time, completely relaxed. It just felt like everything would be OK now.
Then there's the hair tie that Zeniba gives her.

"It will protect you. It's made from the threads your friends wove together."

- These "friends" being a shadow creature called No Face, a formerly spoiled baby, and a bird creature who used to be loyal only to Chihro's nemesis Yubaba. Chihiro's really good at making friends.
If you watch it with subtitles, you can see how much that they changed from the Japanese script to the adaptation. Like Haku bargaining his life in exchange for allowing Chihiro to return home. That's right: if Chihiro hadn't figured out his real name, Haku would have been killed. That enough love for ya?
Three Kamaji moments:
- A small one, but calling Chihiro his granddaughter. Sure, it's a cover story, but his tone of voice, at least in the English dub, sounds so much like a real grandfather introducing his granddaughter to other people.
  - Even better is that before that moment, he wasn't overly fond of Chihiro, insisting that he had no work for her and getting angry when she throws the piece of coal into the furnace (he accuses her of trying to take work from the enchanted soot). And then he just lies so casually, to keep her from getting into trouble with Lin.
- Him covering Chihiro with the pillow when she falls asleep in his basement.
- Giving Chihiro the train tickets that he's been saving for forty years.
It's still rather creepy, but No Face asking to see Chihiro after he has been poisoned by the greed in the bathhouse. Even though he's gone so bad, he still wants to see her and, in his own twisted mind, repay her (by that point, he is mimicking the bathhouse employees and believes that giving Chihiro gold will make her happy).