The tooth pulling scene (in the original) and the 2005 revival.
"Epiphany" is particularly horrifying. Watching Sweeney take a flying leap into the pit of insanity, combined with the song's tendency to switch between his declaring vengeance on all of humanity and mourning his wife and daughter. In some productions, the leap is literal: the actor playing Todd will jump off the stage and start shouting in audience members' faces.
The beggar woman becomes that much more horrifically tragic when you realize she's Sweeney's wife and the combination of the traumatic assault, her attempted suicide, and her time in Bedlam House have left her a shell of her former self.
Judge Turpin pretty much.
Judge Turpin's version of Johanna is sometimes cut...and for good reason. A horror-thriller musical about serial killers and cannibalism is disturbing enough but Turpin flogging himself to get rid of the impure thoughts about his ward and having a frigging orgasm because of it takes it to a whole new level.
Judge Turpin, in a terrifying owl mask, raping Lucy as his group of socialites simply watch an laugh, along with the Beadle making a sick grin. And unlike most performances of the musical, you can hear her screams as everything is going on. It's widely considered to be even scarier than the abundance of violence. "Poor thing", indeed.
In some productions, the shrieking laughter at Lucy's expense are emphasized by the orchestra's string section Mickey Mousing some nasty squeaking noises.
ALL OF THE THROAT SLITTING SCENES! Unlike the stage show, which use clearly harmless stage trickery, this movie loves getting up close to the victim's throats as Todd really digs his razor in, blood splattering everywhere, complete with nauseating sound-effects to make it extra-visceral. The worst is Pirelli's: the pained gurgling noises really sell it.
And then the way they're all dropped onto their skulls via the chute to the cellar. Owtch!
The beggar woman's capacity for this was ramped right up for the movie.
The scene with Toby in the evil basement, when the poor kid finally learns what's in the pies and then has to watch as the just-murdered Beadle Bamford gets dumped right down into the basement with him. And then Mrs. Lovett and Sweeney come down into the sewer looking for him because they want to murder him too.
When Johanna finally comes face to face with Sweeney, he is absolutely covered in blood due to having just savagely murdered the bastard responsible for transporting him for life when she was just a little baby and dropping him down the chute. He doesn't recognize her as his daughter because of her disguise and shows every sign of wanting to kill her for being a witness to the horror he's just unleashed. That's gonna stick with you for a while...
Beadle Bamford. While Judge Turpin raped Benjamin Barker's/Sweeney Todd's wife, the Beadle was looking on with too much interest. Later, he praises Turpin for sentencing a nine-year-old boy to death for stealing a pretty small, insignificant something.
Less severe than other examples (because he's shown to be completely fine a couple scenes later,) but the Beadle's absolutely brutal beating towards poor Anthony is pretty hard to watch. Especially because Anthony is one of the only genuinely kind and sympathetic characters in the film. Beadle even does a little dance-like "skip" with each blow, making it clear how much he's enjoying it.
When Sweeney finds out Lovett lied to him, he takes her off her guard by dancing with her and throws her into her own oven to be burned alive. Unlike the stage play, we can see exactly what happens as she screams and thrashes while she burns up. The last shot of her shows a shrieking, thrashing blackthing, with her face havingmelted off! It's telling that, for a movie with multiple scenes of people getting their throats slashed, many viewers considered this to be the scariest scene in the movie!
And just before that, Sweeney trolls her by tricking her into thinking that he's perfectly fine with this and is as in love with her as she is with him, just so that her fatal punishment will be even worse!
Toby killing Todd. In the play, his Heroic BSoD is shown by having him stumble about while mumbling nursery rhymes with a crazed half-laugh. Here? He crawls out of the sewer, casually picks up Todd's razor, kills him where he stands and lumbers out the door. All while completely stone-faced. There's some indication that his resulting trauma has turned him into another Todd.