After saving Simba from the hyenas and scolding him for endangering both himself and Nala, Mufasa and Simba have a heartwarming father/son moment:
Simba: Dad? We're pals, right?
Simba: And we'll always be together, right?
Mufasa: Simba, let me tell you something my father told me. Look at the stars. The great kings of the past look down on us from those stars.
Mufasa: Yes. So whenever you feel alone, just remember that those kings will always be there to guide you... and so will I.
What makes it even better is that Mufasa diverts Simba's question a little, whilst answering it at the same time. He's effectively telling Simba that he won't always be around, but that he will always be with him, at least in a sense.
Simba expected to be getting the stern disciplinary lecture that no kid wants, and it turns into a father/son moment from there.
Mufasa: Being brave doesn't mean you go looking for trouble.
James Earl Jones performance during the scene, hearing his deep dignified voice when Mufasa laughs while rough housing with Simba somehow makes the whole scene even more adorable.
Plenty of obvious examples in this movie, from some of Mufasa's talks with Simba to the "Can You Feel The Love Tonight" song, but especially worthy of mention is when Timon proves he really is a Jerk with a Heart of Gold he says, "Well Simba, if it's that important to you, we're with you to the end!", because Simba's his friend and they're in this together no matter what.
The end, with the "Circle of Life" reprise playing as Rafiki presents Simba and Nala's new baby to the world, just as he did with Simba himself at the beginning of the movie.
Simba as as a newborn cub in the beginning is just adorable to some.
The audio commentary even mentions that the moment where baby Simba turns around always got an "Awwwww..." whenever they saw the film with an audience.
That moment when Simba feels most alone, then suddenly Rafiki appears, chanting his rhyme, to help him.
Which has the payoff in what he shows Simba in the jungle pool. Here we have Simba, once he finally realizes the "creepy little monkey" isn't completely yanking his chain, but actually did know Mufasa. He tries to break the news of his death to him gently, only to be told, "He's alive, and I'll show him to you!" Despite his misgivings, he follows Rafiki (one of the storybook novelizations even had him think in his thoughts that he actually would get to meet his father again—he's that lonely, desperate, and guilt-stricken he naively believes his father is literally still alive!) until they reach the pool. There he has to again break the news: "That's not my father, that's just my reflection."
Rafiki: No. Look harder. (The reflection changes to that of Mufasa) There, you see? He lives inyou.
Rafiki: Ah yes, the past can hurt. But the way I see it, you can either run from it...or...learn from it.
Rafiki when he discovers Simba's alive. He starts laughing hysterically as he takes red paint and paints a mane on the picture of Simba he smudged when he thought he was dead. Accompanied by awesome music.
Crossed with Crowning Moment of Awesome, Simba ascending Pride Rock in the rain and roaring - how proud everyone was of him, seeing him nuzzle his mother and Nala, and Zazu bowing to him. Considering the grief he gave Zazu during his cubhood...
Fun fact: if you look closely when Zazu bows, you can see his beak moving but no sound. This is because the script initially called for him to say "Your majesty" when Simba walked by, but it was scrapped because they decided it would be more poignant if there was no dialogue. With that in mind, the scene becomes more heartwarming. This is kept in the theatre production.
The entirety of Can You Feel the Love tonight certainly qualifies, but special mention goes to Simba and Nala's reunion. These two are in the middle of an epic beatdown, and all it takes is for Nala to pin Simba for him to remember her. Even though he's been gone from the Pride Lands for years, he never forgot his best friend. Then it goes up to eleven when Nala realizes that the lion she's been fighting is her friend she thought was dead and gone and the two react like...well like cubs.
When Scar and Sarabi are arguing, Sarabi yells at Scar for not being half the king Mufasa was. Scar hits her and knocks her down. This enrages Simba and he goes over to nuzzle his mother.
Two out-of-film examples, proving just how much the film touched audiences around the world:
In a program which aired in the UK back in 2005, detailing the top 100 family movies of all time (TLK was number 6), one of the producers revealed that they received a letter from a young boy who had been devastated at the loss of his father. Unsure what else to do, his mother took him to see the film as a way of cheering him up, completely unaware of what happened in it... But rather than make things worse, the film actually helped him come to terms with what happened.
Another similar story was detailed in the special features of the 2-Disc Special Edition, in which a man lost his wife but had no idea how to tell his young child, who was too young to understand what death was. So he showed his child the film, and then explained that while Mummy was gone, Mummy was always watching over them. The woman telling the story was nearly in tears by the end.
"...and when your films do that...it makes it all worth while."
At the end, as Rafiki beckons Simba towards Pride Rock. He bows to him and Simba grabs him in a tight hug, effectively thanking him for all that he's done for him.
When Sarabi sees a grown up Simba for the first time.
Simba: No. It's me.
Sarabi: Simba? But how...?
Simba: It doesn't matter. I'm home.
Combination this and Crowning Moment Of Awesome when Simba and Nala are being chased by the hyenas in the Elephant Graveyard. When Nala falls behind, Simba immediately turns around and rushes to her defense, scratching the face of the much larger hyena and giving Nala time to climb away. And when they are cornered, he places himself between Nala and the hyenas and tries to roar at them in a desperate attempt to intimidate them. He's just one small cub against three adult predators, but he intends to go down fighting to defend his friend.
Simba and Nala had been nothing but a thorn in his side for the entire day and when he had finally caught up to them after they ditched him, he is understandably furious and irritated at the disrespect and disobedience of his charges. But the moment the hyenas reveal themselves, he immediately raises his wings and shields the cubs. He is on the ground the entire exchange and makes no move to fly away until the hyenas are thoroughly distracted and the children are running.
When Shenzi stops them from leaving and hints at "having them for dinner", there's a quick blink and you'll miss it moment where Zazu, who is only half the height of the cubs, places himself in front of the children with his wings outstretched and closes his eyes, fully prepared to die for his prince. Zazu may be a stick in the mud and even have traits of a coward, but would do everything in his power to serve and protect the royal family.