Factual error: When Rose is locked in the room and the bodies come toward her, you can see a modern-day electric light-switch to the side of the door she's trying to get through. When the Doctor hears her screaming for help and runs for her, you can see a central heating radiator. In fact, the undertaker's house has at least two radiators in different hallways. Both are wildly anachronistic for the time period. (00:16:30 - 00:17:08)
Suggested correction: While the light switch is ahead of its time, the heating radiator was invented by Franz San Galli in 1855 and was immediately popular. It's possible that by 1869, when the episode takes place, that Mr. Sneed would have one.
Other mistake: At the end, the Doctor and Rose treat Charles Dickens by making the TARDIS dematerialise in front of him. The engines begin to grind, but the time rotor (the thing in the console's central column that goes up and down when the TARDIS is flying) does not move. (00:42:53)
Continuity mistake: Early on, when Rose is talking to the Doctor about how amazing it is that they can visit days that only happened once, her hair moves from tucked behind her ear to covering it, then back to its first position. (00:05:00)
