Continuity mistake: When Melinda Dillon breaks the lamp, it is broken into many pieces, but when Darren McGavin is gluing it back together it is now in much fewer and bigger broken pieces. Obviously different broken lamps were used.
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.