Snow White is making a pie for Grumpy right before the Queen shows up.
After the chase and death of the Queen, the dwarfs return home to find Snow White dead and the soundtrack skips to what seems like complete and total silence for a full minute; after the musical chaos of the previous scene it's the cinematic equivalent of a punch in the gut. There's actually very soft organ music playing in the background, but the effect is still there. The shots of the Dwarves crying... ANY shot of a Disney character crying is pretty saddening, especially Grumpy, who is far in the corner of the room they're in, and begins to cry just as well.
Grumpy can't even bear to look at her after all the crap he gave her. He had to have been thinking "I was nothing but mean to her, and I never got to tell her that I did like her."
There are stories of how Snow White was allegedly the first animated anything to make an audience cry at the premiere. After months prior to the release of people calling the movie "Disney's Folly," saying that an hour and a half of color/animation would drive people mad, etc., a bunch of grown celebrities/film industry members broke down over the sight of a dead cartoon character. Seeing Dopey sob into Doc's shoulder probably didn't help...
Probably the biggest reason this scene was such an effective tearjerker is because other than the mourning animals, the scene was played completely straight and realistically, with the exact same atmosphere one would expect at a real-life funeral. No epic soundtrack, no great meaningful speech, no symbolic magical events... just sadness and loss.
When the dwarves are surrounding Snow White's coffin and are all placing flowers, Grumpy is the one who goes up to put the flowers into Snow White's hands.
After the Prince kisses Snow White, he kneels by her coffin and lowers his head in despair. He had no idea his kiss would wake her up.
This doubles as a moment of heartwarming as well, but the scene with the Huntsman when he hesitates to kill Snow White, you can't help but feel bad for him. He didn't want to do it, but he knew it would be his death if he didn't. Regardless he made her run for it instead.
The reason he has an opportunity to kill her in the first place is because she's attending to a lost baby bird and thus isn't paying attention to him. The juxtaposition of her kindness and what he's trying to do creates intense suspense, tears, and from there relief.
Following this he breaks down and begs for the princess' forgiveness, whose reaction is mere confusion, forcing the Huntsman to desperately Shoo the Dog.
The most Tear Jerking/Heartwarming thing about this movie is not actually a part of the story. In Don Bluth's guide to animation, Bluth himself, former Disney employee, described the point during production where Disney ran out of money halfway through the film. With no way to finish, he had to go to the bank and beg for another loan in front of a board of executives. As an aid, he brought along fragments of the half finished film, fragments which astounded the entire board. When finished, the Head Executive got to his feet and made what, for a business man at that time especially, constitutes a tremendous leap of faith into the complete unknown:
"Gentlemen, in fifty years time, nobody will remember the names of any of the people in this room. But they will remember the name Walt Disney. I'm in."
The entire final scene, reawakening included.
"Someday My Prince Will Come." Why? The thought of someone wanting to harm Snow White-a young, gentle person-is heartbreaking.