Factual error: When Sweeney is sharpening his blades, most noticeably during the shaving contest, he is sharpening toward the blade, which you simply can't do with a razor blade and leather strop. A razor will slice right into the leather. He also doesn't even have the blade flat to the leather so again they're not remotely sharp as a sharp blade would easily sink right into the leather. Pirelli is using roughly the correct technique, even though they're trying to portray him as sloppy. (00:35:50)
Other mistake: At the end of the song "Wait," when Sweeney and Lovett hear Anthony running up the stairs, Sweeney runs behind the door to his shop. For a quarter second, as Sweeney places his back against the wall, there is a modern-day water bottle on the window ledge. This, however, cannot be seen on the DVD or Xbox, though on laptops and probably computers also, when streaming you can see it around 00:43:10. (00:43:10)
Continuity mistake: Just after Lovett finishes singing "Wait", Anthony comes running up the stairs and Sweeney hides behind the door holding the razor in his right hand. Anthony turns to Sweeney to speak, and Sweeney places his right hand on Anthony's back and guides him toward the window. You get a glimpse of the closed razor. The camera instantly changes to looking at Anthony's back,and Sweeney's hand is empty and fingers are spread out. When the camera angle changes again and Sweeney sits Anthony down, you get a quick glimpse of the razor in Sweeney's right hand. (01:13:00)
Other mistake: Based on the location of the barber's chair with the picture window in Todd's barber shop and the layout of the exterior of the building, there is no possible way that the chute behind the chair would be able to dump the bodies directly into the basement without going through the center of Ms. Lovett's shop.
Continuity mistake: While Sweeney is upstairs with Pirelli, Mrs. Lovett gives Toby a meatpie downstairs. On a closeup, half the pie is gone, but when the shot widens right after, only a couple nibbles are gone from the cake.
Continuity mistake: Pirelli throws his cape and it leans smoothly on the seating part. From the wide angle, half a second later, it is wrinkled on the seating part.
Continuity mistake: The blood on the window, when Sweeney kills Turpin, keeps appearing and disappearing all the time.
Continuity mistake: When Mrs. Lovett has given Sweeney a glass of gin (when they are deciding what to do with the bodies), he takes a large swig and another smaller one and the gin is nearly gone. When it cuts away to Mrs. Lovett and back to Sweeney, the glass is full again and there was no sound of another glass being poured.
Continuity mistake: Nearing the end of the song "Pretty Women," Sweeney is bringing his razor closer and closer to the Judge's neck (right around the line "proof of heaven as you're living"). At one point, he brings the razor right up against to the Judge's neck. In the next shot, the razor is a few inches away from the neck. In the shot following that, the razor is back up against the skin.
Continuity mistake: When Todd is first shaving Judge Turpin, he places the barber's brush in the shaving mug twice - once before he turns away from the barber's chair, and again in the next scene as we see him move toward the camera.
Continuity mistake: During "The worst pies in London scene" the wooden rolling pin keeps moving around between shots.
Continuity mistake: When Anthony first arrives at Judge Turpin's house, he sits on the bench across the street. This bench is positioned on the sidewalk between the iron gate's two large stone pillars. During this scene, the position of the actual bench changes, moving noticeably closer to either the left or right pillar, depending on the given camera angle.
Continuity mistake: When Sweeney Todd and Mrs. Lovett are dancing in her storefront (after they finalize their plan of what to do with the bodies), Depp is holding a butcher's cleaver in one hand. In the wide shots of them dancing, the cleaver is noticeably lower than in the close ups, where the cleaver appears to be at the level of Mrs. Lovett's shoulder.
Continuity mistake: While Anthony sings "I feel you", the passersby behind have no continuity whatsoever.
Continuity mistake: Toby is introducing Pirelli and his fringe and hair around the ears keeps changing style all the time.
Continuity mistake: While Pirelli sings "He take the", after a shot on Sweeney's blade, Pirelli's hand swaps from being on the man's forehead to the chin.
Continuity mistake: The first time Judge Turpin is in Todd's chair, the shaving cream changes during this scene. When Todd lathers him up, covering Turpin's entire lower face with shaving cream, including covering the lips with a large blob of foam, but in Judge Turpin's next closeup, however, not only is the blob gone, but there is a streak at the corner of his mouth where it looks like someone wiped it away.
Continuity mistake: While Sweeney sings "I'll Never See Johanna," Ms. Lovett is sitting on the chair with him behind. In the close-ups, they're close together; in the wide angles there's a big space between them.
Continuity mistake: The way Toby holds the elixir bottle while the baldie is rubbing it on his head changes in a split second from angle to angle.
Continuity mistake: When Sweeney is showing Mrs. Lovett the razors right before they start singing, he pulls one out and opens it out straight. He stands up and the blade is still straight. It cuts to a closer shot, and the razor is at a noticeably different angle, then it goes back out to a full shot and the razor is straight out again.
Answer: From what I heard, the chorus songs (known as "The Ballad of Sweeney Todd") were cut due to time and inability to make the chorus work on-screen. Actors were set up to play "ghost narrators" at first, so that indicates that the filmmakers did want to include the songs. They were just not able to find a way to make them work satisfactorily for them. Also, I think (can't verify) I also heard that Stephen Sondheim, the show's original creator and composer/lyricist, agreed that these cuts needed to be made for the film to work.