Mufasa's demise, obviously. The scene where Mufasa is killed shows the body pulled about by his son.
It's especially sad for kids because if something so horrible could happen to the perfect father, who's to say it couldn't happen to their own father?
The music that plays during the scene certainly doesn't help, especially when the part when Mufasa's theme comes in (fittingly it was based on parts of Mozart's Requiem Mass). Add Simba trying in vain to wake his dad up, and then in utter despair trying to cuddle up under his paw, and you have a scene that is possibly one of the saddest in any animated film ever.
Simba's Big "NO!" moment right as Mufasa dies could also qualify, since it's pretty much a given what's going to happen at that point.
The scene also became a tearjerker for the film's creators, as they received a letter from a young boy who had lost his father, thanking the creators for the film as it helped him deal with his loss. The staff were moved to tears by it.
The scene is even more heartbreaking in theaters, in 3D...
This scene is arguably worse than the death of Bambi's mother. For one thing, there isn't any Mood Whiplash to this scene. Afterwards, Scar tells HIS OWN NEPHEW that it's his fault his father is dead, to run away and never return, before sending the Hyenas kill him. There's a little bit of comedy courtesy of Banzai falling into a bramble patch ("Cactus Butt!"). Plus, unlike Bambi, where we didn't actually see his mother's death, we got treated to the visual of Mufasa getting pulled off the cliff and falling all the way down into the stampede.
There's also the fact that, despite being a film character, Mufasa is a real character with heart and humor who the audience has come to love and sympathize with. To see him snuffed out so cruelly is heartbreaking for audiences.
Afterwards, however. we see Scar giving a none-too-sincere eulogy for Simba and Mufasa just before he takes the helm, and we see the lionesses and Zazu's reaction to this, with Zazu comforting the heartbroken and shattered Sarabi... And then the hyenas come out. And these aren't the goofy trio that we know and love; these hyenas are Nightmare Fuel.
Followed by Rafiki's reaction, and then Simba almost dead in the desert. A lot of people probably consider it a good thing Timon and Pumbaa showed up when they did.
Two Meta examples with voice actors. Jonathan Taylor Thomas was recording this scene and his mom was present so his director told him to imagine his mother falling; he proceeded to screw up the take by yelling "MOM!" instead of the scripted "DAD!" And then in a different dub (I can't quite remember which), there was a break in the recording and the director noticed that Simba's cub voice actor was unusually quiet and asked what was wrong. Turns out the kid's father had recently died.
The size of this point proves that this scene is, and will remain, within the top 5 saddest moments ever in animation history.
Just before this, when Simba is at his lowest point since the stampede, all of his feelings of guilt and shame having been brought back by his argument with Nala. He starts off in what seems like Calling the Old Man Out/Rage Against the Heavens, but it quickly turns once again into self-blame. It's definitely a moment where Matthew Broderick proves He Really Can Act.
Simba: You said you'd always be there for me!! But you're not... (beat) And it's because of me. It's my fault...it's my...fault... (chokes up and starts to sob)
The ending when Simba takes the throne is one of these too, partly from Tears of Joy (and the intense emotional power of Zimmer's score) but also just the sense of sorrow for everything Simba went through to get to that point and the relief that he is finally there, he's earned his place and honored his father. One word sums it up: "Remember."
Although not quite as bad as the other tearjerkers, when Mufasa saves Simba and Nala from the hyenas you feel Mufasa's distress at his son being in grave danger if you're a parent. Driven home in the following scene when Mufasa confesses to his son that the possibility of losing him is the one thing that made him genuinely scared. Even moreso during the stampede, when you can clearly see the distress on Mufasa's face as he searches desperately for Simba among the mass of wildebeests.
Every parent who genuinely loves their kids will instantly sympathize with Mufasa on this. Imagine your son and his close friend go out to the place you explicitly tell him not to, knowing full well that the place they're going to is VERY dangerous, without any parental supervision to boot. And then three creepy strangers start chasing them about, with God know what kind of intentions. Virtually any parent is suitably freaked the hell out at the idea.
There's a very understated bit during the stampede. Zazu is flying through the gorge searching for Simba and finds him desperately clinging to a tree branch over the rampaging wildebeest. Simba cries out for Zazu to help him, and all Zazu can do is reassure him that his father is coming and tell him to hold on. Simba has caused Zazu plenty of irritation in the past, but Rowan Atkinson's excellent voice acting definitely gives the impression that right now Zazu would give anything to be able to pull Simba out of harm's way. But he knows he's not strong enough to lift Simba and therefore can't save him. That probably haunted him for years.
Nasty Fridge Horror when considering Zazu for this whole thing. After Mufasa goes down into the gorge to save Simba, Zazu tries to fly back to Pride Rock and get help, only to be smacked into a wall by Scar and knocked unconscious. Imagine what happened when Zazu woke up. He probably thought he panicked and hit the wall himself and failed to get potential help for Mufasa and Simba because of it. Until the truth comes out later, it's not unreasonable to think that he blamed himself and felt that he failed both Simba and his king. It makes him seem very woobieish.
Some more fridge horror for that scene; how do we know Scar didn't pull the "it's all your fault" trick on Zazu, too?
The look of utter horror and betrayal on Mufasa's face when he realizes his own brother is about to throw him to his death. In spite of the vitriol shown between Scar and Mufasa at the beginning, Mufasa is still a good and noble character and he probably still cares about his brother and is completely unaware of Scar's hatred for him until the very end. Someone he knew as family, someone he probably loved and wanted to see happy, tricked and brutally murdered him.
The hyenas looking back at Scar, appealing for food.
The stage musical
Although a great deal of The Lion King is deeply emotional and affecting (as is the film it was based on), there's one particular addition not in the movie: the solo number performed by Simba, "Endless Night". In particular the lines: "You promised you'd be there/Whenever I needed you/Whenever I call your name/You're not anywhere/I'm trying to hold on/Just waiting to hear your voice/One word, just a word will do/To end this nightmare." Also, the earlier lines "Sleepless, I dream of the day/When you were by my side/Guiding my path/Father, I can't find the way." By the time the end of the song comes, Simba sings such a stirring, heartfelt testimonial to hope for the future, but the way the song uses actual lines from the movie and builds upon them to reference both the deep love Simba and Mufasa shared (and how he was such a perfect father) and the guilt, despair, and suffering Simba went through as mentioned above is just so overpowering and gut-wrenching.