The Ship Tease between Ash and Misty is so obvious. Melody has been ragging Misty for the entire movie about Ash being Misty's boyfriend. Finally Misty decides it's time to go out into the apocalyptic storm to rescue her man. "Ash is never really alone, because he has me."
It doesn't even need the Shippy elements to work, because even without them you can tell these two are True Companions. Before the Original Series ended, Misty was the human friend who stuck by Ash since Day One (up until their long journey together was finally over), and traveled with him through more regions than her "successors". It's small wonder the audience is attached to the dynamic of the original travelling group. Say what you will about shipping or whatever, this scene defined the relationship between Ash and Misty.
Misty and Melody become "official" friends at that point too, which is doubly heartwarming.
Ash and his mother near the end of the movie. Delia scolds him for taking such a risk with his life, then explains that while he was saving the world, she came so close to losing her whole world. The whole exchange is just a big ball of heartwarming.
Delia: And just remember. Every day...you're my hero.
The little conversation among Jessie, James, and Meowth as they plummet to what they think is their death after jumping off of Lugia to allow it to fly faster.
Jessie: Think we did the right thing? James: I know we did the right thing, Jessie. And it feels great. Meowth: Well, this is it then... (All three move from holding hands into a group hug) Jessie: Let's not say goodbye. James: Let's just say... Meowth: ...we're gonna die.
After they jump of Lugia, Ash calls out to them in an anguished voice. They may be his enemies, but he recognised their sacrifice and didn't want them to die.
As Ash is being flown to do his duty as The Chosen One, he sees all of the Pokemon who have come to the Orange Islands to help and asks what they can do. Lugia says "They are not sure. And that is why they are here." Every Pokemon that could went to the source of the disaster on the off chance that they could do anything to help save the world and any Pokemon that couldn't got as close as they could because they understand that every person can and will make a difference.
Lugia's farewell line: "The Beast of the Sea has been tamed. The fate of the world could not have been in better hands."
The three titans flying together in perfect harmony after the day is finally saved.
A minor one near the start: when the boat hits the rough weather, Tracey's warrior Scyther reaches out to steady Venonat. Seems simple, but Scyther is both the newest addition to the team and a notoriously violent species.
All of Ash's Pokemon coming out of their Pokeballs to offer him their support when he starts to doubt himself.
A minor thing, if you look closely at the end, when the ice melted, you can see the water Pokemon carrying Pokemon who had to use the ice to reach the Orange Island, who would mostly likely be stranded when the ice melts, home on their backs.