After nearly getting himself killed for stealing bread, Aladdin hands the whole loaf — probably all the food he would have had that day — over to two orphans. They don't mean anything to him, they're just two more street rats, two out of dozens... and he hardly gives it a second thought. The look on the little girl's face is just so touching.
And then when the kids begin eating it, we see the girl has given the bulk of it to her little brother, taking only a bite or two for herself.
Also, Abu - he was very reluctant to part with the share of the loaf he'd gotten before they saw the kids, and defiantly takes a big bite out of it... but his conscience gets the better of him, and he gives the rest of his piece to the children after Aladdin gave his. And though he was grudging about it, he brightens up when the kids show their appreciation towards this unexpected generosity (they were incredulous enough that Aladdin gave them his bread since life as a street rat is a struggle). The scene shows not only Aladdin's selfless compassion, but that Abu has a heart of gold beneath his own capacity for greed (for further heartwarming, imagine that Abu represents Aladdin's baser instincts - the kindness that cannot be extinguished even by a life of desperation and need).
The magic carpet ride, one of the most romantic and magical falling-in-love montages ever. That it comes on the heels of Aladdin realizing that his faux charm isn't going to work on Jasmine and reverting back to his natural self makes it even more heartwarming, since the carpet ride is when he knowingly treats her to the one thing she's always wanted: freedom.
"No one to tell us no, or where to go, or say we're only dreaming..."
The cutest part? The fact that Jasmine's eyes are wide as she stares around at the world she's never seen before, but Aladdin hardly ever takes his eyes off her.
"I choose you, Aladdin." It's a perfect line of dialogue that brings both Jasmine and Aladdin's desires full circle: she gets the freedom to choose, he's loved for being himself, and there's nothing preventing them from being together. And then they bop heads. Awww!
Aladdin:(chuckles) Call me Al...
This didn't strike me until I was an adult but just how far the Genie is willing to go for Aladdin just because he was his friend. When he sees Aladdin and Jasmine together at the end, he tells Aladdin that he can use his last wish to become a prince again even though it means he might not ever get a chance to be free again.
Aladdin: But Genie, what about your freedom?
Genie: Hey, it's only an eternity of servitude. This is love! Al, you're not going to find another girl like her in a million years. Believe me, I know, I've looked.
Aladdin wishing Genie free, despite Genie saying that he's willing to stay in the lamp just so Jasmine and Aladdin can be together.
And how about when Genie gets ready to use the final wish to make Aladdin a prince, and is completely caught off-guard when Aladdin wishes for Genie's freedom?
After the Genie has been freed:
Genie: Oh, does that feel good! I'm free! I'm free at last! I'm getting outta here! I'm off to see the world! I—
Aladdin: Genie... I'm really gonna miss you...
Genie: (tears up) Me too, Al. No matter what anybody says, you'll always be a prince to me.
Genie celebrating as he leaves to enjoy his freedom:
"Hey, rugman! Ciao! I'm history! No, I'm mythology! Oh, I don't care, I'm free!" (elated laughter as he flies into the distance)
A meta example happened while the film was still in production. Robin Williams would often send or receive calls to and from Steven Spielberg, who at the time was working on Schindler's List where a lot of the cast was feeling down. Williams would be put on speaker phone and would tell jokes to the cast and crew (including Spielberg) in order to cheer them up.
Aladdin describing why Jasmine is so important that he deserves an exemption to Genie's "I can't make anybody fall in love" rule:
Aladdin: But Genie, she's smart and fun and...
Aladdin: Beautiful! She's got these eyes that just...and this hair, wow...and her smile...ahhh...
Seriously, plain text can't do Aladdin's emotions justice. Just watch this video!
Aladdin telling Jasmine that she stood out in the marketplace. He's so sincere when she says it, and she's so sincere when she smiles back. It's chemistry like this that makes the Fourth Date Marriage (if there only had been the one movie, and they got married at the end) just a little bit more plausible.
A smaller one, but the Sultan, who is SO set on his daughter getting a princely husband, refuses to force her to marry any of the suitors she refused. Even with all his strictness, he loves his daughter enough to not want to force her to marry someone she hates.
Another meta example: any airings of Aladdin after August 11th, 2014 included this little bit between 'The End' and the credits: "In memory of Robin Williams, who made us all laugh."
So far, as of then, any time it's aired it's been uninterrupted.
Also from "The King Of Thieves" after Aladdin and Jasmine have finally gotten married the crowd parts and standing at the back of the wedding pavilion stands Cassim. "Even a wanted man can risk a bit to see his own son's wedding."
You don't think about it much, but in The King Of Thieves, Genie is Aladdin's Best Man at a remarkably western wedding. That coveted job traditionally goes to the best, most trusted, deepest friend of the Groom. Awww...
Also, when Aladdin tells Genie about his Disappeared Dad and his own worries about married life...
Aladdin: What if I'm no good at it? Genie, if my father were here...
Genie: Al, little buddy, if your father were here, he'd be as proud of you as I am.
This line from Jasmine to Aladdin as he's having pre-wedding worries: "People like you just don't come out of thin air."
Iago and Cassim's developing friendship during the course of the third movie — true, they're both amoral scoundrels who initially bond simply over the prospect of getting rich beyond their wildest dreams, but they seem to genuinely like one another, and the ending, where Iago (who over the course of the TV series has been The Drag-Along) willingly goes off with Cassim for more adventure, and Cassim welcomes him along, is genuinely touching.
Cassim: Well, you little turkey? Have you come to say goodbye?
Iago: Nah, I already said my goodbyes. 'Course, the monkey had to cry... (Starts tearing up) ...And that got me goin'...
The Book Ends ending, which is pretty touching on part of the development team; they remembered a one-shot character who only appeared for a few minutes in the beginning of the first movie, giving the series some serious closure.
When Jasmine is tricked into using a magical lotion (by playing on her insecurities concerning her appearance and how much Aladdin loves her), she wakes up to find herself turning into a giant snake. Aladdin and the gang accompany her to find a cure, but when the fruit withers in her hand, she demands that they all return to Agrabah without her. Aladdin then uses the lotion on himself, turning himself into a snake, so that he can stay with her. Awww.
Eden makes friends with Dhandi, a homeless girl, even encouraging her to make better wishes. Dhandi's last wish is for them to be together forever. D'aww. Eden doesn't seem to mind terribly much. Unfortunately, it means Eden and Genie can't be together until Dhandi dies. Still, Genie seems willing to wait.
Basically, any romantic interaction between Aladdin and Jasmine.