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.
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.
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.
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.
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.
The Lion King II: Simba's Pride
The entirety of "Love Will Find A Way", but especially the moment when Kiara and Kovu run towards each other and Kiara buries her head in his mane as the camera circles around them.
The lyrics alone of "Love Will Find A Way" can be this, combined with a serious Tear Jerker.
It's, perhaps, especially poignant for the LGBT community. ("They can have the world; we'll create our own..." "...and if only they could feel it too, the happiness I feel with you...")
In The Lion King 1½, it is suggested that even though Simba's life with Timon and Pumbaa was considered reckless and carefree, Timon and Pumbaa were still caring guardians to him, especially during Simba's cubhood when he was still coping with the loss of his father.
Timon, after years of taking for granted that he had all he always wanted, spends a day alone in his "dream home", only for Rafiki to show up and coax out of him that his real rest and joy was with his friends.
At the end, Timon and Pumbaa bow before Simba before his coronation. And Simba gives them a warm hug in return while what he says next is even better: