Continuity mistake: When Scrooge walks into the Cratchit home at the end, there is a "Home Sweet Home" plaque on the wall. The shot then cuts away from it. The next time the plaque is visible, its lettering has changed.
Continuity mistake: When Scrooge wakes on Christmas morning, he puts his foot through his hat. It has a huge hole in the top when he leaves home, but when he puts it on Tiny Tim's head, it's undamaged.
Continuity mistake: When Scrooge opens the door to the Counting House, you can see that there is a bell by the door, but when we see Scrooge entering the Counting House in the next shot, the bell has disappeared.
Continuity mistake: When Jiminy Cricket (The Ghost of Christmas Past) jumps on to Scrooge's bedside table, the time on the alarm clock reads 2:00. In the next shot, when the curtains move and Scrooge hears the dinging, the time reads 1:00, and when we return to Jiminy Cricket in the next shot, the time reads 2:00.