Plot hole: The guy working for the Daleks offers the homeless men in Hooverville a dollar a day to go into the sewer because the Daleks need more people. Most of the people refused and one called it slave wages. If they were not getting paid until they came back, and since the Daleks were planning on experimenting on them, the people weren't going to come back; why not offer them a large amount of money big enough to entice them but small enough not to be suspicious?
Continuity mistake: When Martha is searching for how the Daleks changed the design of the Empire State Building, she is examining the plans for the top of the tower. In the first shot where the new plans are visible, the upper right shows a more detailed picture of a plate of Dalekanium. Seconds later, when we can see the new plans again, it's the same side view but the enlarged picture of the plate with the Dalek bumps is gone. These are still the same new plans (the old plans are on her right) and there was no time or reason for Martha to take another page. (00:24:40)
Other mistake: After the hybrid Dalek Sec explains the Daleks' plan to harness the power of a solar flare to the Doctor, when the Doctor asks how many minutes until the flare arrives, Sec says it's in 11 minutes. A fair amount of action and backstabbing later, when the Doctor and Laszlo arrive at the top floor of the Empire State Building and after the Daleks take back control of the elevator, the Doctor says that the solar flare arrives in 6 minutes. It is a considerable stretch for everything that happens between those two moments to take place in only five minutes.
Continuity mistake: After Martha hugs her mother, when Francine asks what the occasion is and Martha responds that she's just pleased to see her, in a shot facing Francine she has her right hand on the side of Martha's head. In the next shot, facing Martha, Francine is moving her hand to the position it was in the previous shot. (00:06:15)
Continuity mistake: When Lazarus starts his speech, there is a shot of the Doctor and Martha watching. The Doctor has his arms at his sides. The scene cuts to Lazarus speaking, a shot in which the Doctor is visible from behind, unmoving. In the next shot of the Doctor, he has his arms crossed. (00:07:15)
Continuity mistake: During the reception, the newly rejuvenated Lazarus takes a tray of hors d'oeuvres from a waiter and starts eating them while talking to Lady Thaw, just as the Doctor and Martha walk up. Later in the conversation, just as Lazarus says to the Doctor that what happened inside the capsule was exactly what was supposed to, the tray vanishes. (00:11:10)
Continuity mistake: When Francine calls Tish over to ask her if she's seen Martha and the Doctor anywhere, Leo is taking a drink from a nearly-empty glass of champagne. When Leo is next seen on camera, as Francine asks Tish if Martha had ever mentioned the Doctor to her before, he's drinking from a much fuller glass of champagne. (00:16:00)