"Can you see me?" For the first time in 300 years, someone other than his fellow myths can see, hear, and interact with Jack.
Just how excited and happy Jack is during that scene. He's breathless, his face is lit up, and he's jumping around, grinning. We haven't seen him like that at all the whole movie. His eyes even look like they're about to tear up. That moment is among the most enlightening in the film.
From just before that, Jack restoring Jamie's faith in Bunnymund. Note that Jack and Bunnymund have been at odds the whole movie. Watching Jack selflessly act on Bunnymund's behalf, knowing that Jamie cannot see him will bring a tear to your eye.
And Bunnymund's reaction later is definitely heartwarming. Jamie is initially quite surprised at Bunny's diminished form, leading Bunnymund to assume that Jack told the boy to point it out just to be a punk. But when Jamie explains that Jack is the reason Jamie still believes in the Easter Bunny, Bunnymund is speechless, touched, and honestly rather tearful.
Leads to a bit of fridge logic when you realize that there was a human that Jack interacted with: Jamie's sister, when he carried her home.
She's a really young kid who is at an age that she'll believe in anything. Not to mention she was outside when her mother made a reference to "Jack Frost nipping at your nose" and could have easily heard it. If one of the other Guardians said someone named Jack was going to take her home, she'd believe that someone was going to do that.
There's also just after Bunnymund discovers Jamie's toddler sister, Sophie, in his domain and initially treats the intrusion as an alarming embarrassment. That soon passes with Bunnymund gently pushed to welcome the little girl to his world and participate in his egg painting with a warm smile.
Double heartwarming points for Jack being the one to push him with his magic, which apparently influences people to get along and have fun.
Anything to do with little Sophie really, as all the guardians are reminded what its like to actually interact with a child again.
This is also a poignant moment for Jack that bluntly foreshadows his inner nature and ascension to Guardian status in just the right way; when he figures out that none of the current Guardians really interact with the children under their charge. North and Bunnymund both only make their presence known once a year while Tooth and Sandy - although operating every day...do it at a distance and en masse. Jack calls them out on it, and right then, you see that the children genuinely matter to him and some part of him knows his place with his charges.
The adorable way Sophie clings to Jack as he tries to put her in bed. Not to mention the way she just curls up in Bunnymund's arms and goes to sleep.
For a beat at the end, after Pitch is defeated and he watches the Guardians and children throwing snowballs, you can seen Bunnymund hopping over to little Sophie and helping her make a snowball of her own. D'aww...
Bunnymund telling Sophie after they have defeated Pitch that he's going to miss her, and she gives him a hug.
There is the ending with Bunnymund warmly welcoming Jack's induction as a Guardian, having forgiven the boy for his mischief and earned his respect.
Bunny: That's my boy.
Early in the film, Jack helps the kids befriend a large and intimidating girl named Cupcake. After the snowball scene, she is always seen with the other kids, showing that she has become part of their True Companions. Despite what the other Guardians initially think about Jack, he has always watched and protected children.
Cupcake in general is pretty heartwarming, especially for people who were that bigger-than-average, intimidating kid growing up. Especially her brief moment in the climax, when she changes her stolen dream back from being a Nightmare and snuggles with the resulting unicorn. It's just...gah! So cute!
At the end of the movie, Jamie runs up and gives Jack a hug. Jack is frozen in surprise for a moment before he smiles and returns it. It's literally the first time something like that has happened, and you can tell it means a lot.
The very first scene, where Jack finds his staff and is experimenting with his powers for the first time. It turns into a Tear Jerker soon after, but he just looks so happy and adorable at that moment.
It's his excited bouncing (reminiscent of Rapunzel's excited bouncing) that can get to you.
If you pay attention later you'll notice that his sister does it, too. Watch carefully in one of Jack's memories, when he has the antlers. Adorable.
For all his griping against Jack, it is noteworthy that when Pitch says Jack must be used to being ignored, Bunnymund immediately tries to attack him.
It's hard too tell if it's the fact that Bunnymund made a similar low blow earlier (possibly an hour or less earlier, really) makes this more or less heartwarming.
Relatively small one, but in the ending credit run, all the helpers; the mini fairies, the eggs, the elves and the yetis are all helping the sleeping children back home- trying to cover for them. The capper? When Phil notices Jamie has a book on his kind and signs it as a gesture to believe.
North immediately volunteering the other Guardians to collect teeth in lieu of Tooth's fairies to make sure the children don't stop believing in her. Plus the boys just get so into it.
The giant snowball fight when the Guardians and the children have defeated Pitch. Made even better by Sandman's dream sand creations floating through the background.
North winking at Jack just before getting smacked by a snowball in the back of the head. His fatherly relationship with Jack is another element of heartwarming by itself.
Sandman's return. He creates a hat out of sand and tips it to the children... and Jamie salutes him back. Plus the other Guardians' joy over his revival is adorable.
Jack is still a bit iffy about being a Guardian, but when they arrive at the Tooth Fairy's palace and it's being attacked by Pitch's minions, he immediately goes to help them. Actually, his entire relationship with Baby Tooth is unbelievably sweet- when Pitch demands Jack hand over his staff in exchange for Baby Tooth (and is quite clearly hurting her in his hand), Jack doesn't even hesitate before handing it over for her.
Jack saving his sister. Doubles as a massive Tear Jerker, especially considering that she watched him die.
Bunnymund standing in front of the children and telling him he'll protect them in the climax...despite being the size of an ordinary rabbit at that point.
For that matter, Jamie stepping forward and saying that he'll protect the guardians... followed by Cupcake and the rest of their friends stepping up, too.
The Guardians' expressions of sympathy when Pitch despairs over not being believed in. Despite him being an utter bastard who would kill them all if he could and almost destroyed everything they've worked for, they still feel sorrow for his suffering.
The moment with Jamie and his family after the sled scene. Of particular note is his dog, who jumps on Jamie and starts licking his face at the end.
The weird friendship that develops between one of Santa's elves and an Easter egg. If you watch during the ending, the elf snatches the egg out of the way of Santa's sleigh.
Something about Bunny being the one to raise everyone's spirits by reminding them that Easter is the next day and that it's their chance to restore the lights gets me right in the heart. Up until this point in the film, he's ricocheted between grouchy and arrogant, but his rallying speech shows just why he's the Guardian of Hope.
The friendship between Jack and Sandman. He's openly antagonistic towards Bunny, he's tried repeatedly to break into North's workshop, and he's barely even heard about the tooth fairy, much less met her. But he smiles when he sees Sandy spreading his dreams, he looks to Sandman for answers when the Guardians kidnap him, and is unfailingly polite even when the Sandman can't properly explain what's happening. The Sandman breaks up Jack's fight with Bunny. Jack and Sandy share a look before Jack trips the alarm clock in Jamie's bedroom. When he sees a Nightmare immediately after, he asks Sandy to help him even though he's never asked help from anyone and Sandy does help, even though he wants to wait for his fellow Guardians to recover. All these little moments, but Sandman and Jack are just unfailingly warm to each other throughout the entire film.
One of the prequel comics involves Monty (the boy with glasses) having a sleep over at Jamie's house. Earlier Jamie had almost seen Jack while playing pretend and ended up finding some of his hair (Jamie thought that it was Bigfoot's) and put it into a book of stuff like that. He then tells Monty about various close encounters he'd had with Tooth, North (he lost a tooth on Christmas) and Bunny. Specifically, he almost saw North and took a picture of Bunny's eggs' footprints. He thinks that Monty will think that it's weird, but Monty turns out to have a story of his own. The previous fall he had been attacked by a rabid dog when he was cornered at a lake. To his surprise, the ice froze over creating a safe path away from the dog for him. It then turns out that Jack had saved Monty and was listening in on Jamie's story; before he leaves he leaves a little frost on the windows and Jamie decides to add a new picture to his book. Did I mention that the entire story is available on the wiki?
A meta example. Fans of the movie love it so much that they've made a project for a book called Book of Belief. The goal is to send copies of it to the Dreamworks Animation Studios in hopes of inspiring them to create a sequel to Rise of the Guardians. Search "Book of Belief" in the Tumblr search engine and just look at how many entries there are.