Revealing mistake: When Rizzo falls down the chimney and lands on the goose at Bob Cratchit's house, he burns his feet. When he is next to Gonzo, his feet are smoking. If you look closely, you can see the holes in the puppet's feet where the smoke comes out.
Revealing mistake: During the ghost of Christmas present, Kermit is walking with tiny Tim on his shoulder and you can tell Kermit is walking on a spinning drum. Right before the camera pans away you can see Kermit's feet slide like he stopped walking but the road kept spinning. (00:55:00)
Other mistake: Near the beginning, Scrooge leaves Scrooge & Marley and walks home. As a result Bob Cratchit and the book keepers clean up and Bob Cratchit locks up with a key. Yet the next day when Scrooge gives gifts to the other book keepers, not only is Scrooge able to get inside without a key, the door is unlocked as well. (00:16:45 - 01:16:05)
Continuity mistake: When the song about Ebeneezer Scrooge is about to end, just as Ebeneezer is walking down the last lane, the first Muppet you'll see to the left when he has come down the stairs is wearing a red weird hat. Just a second after Scrooge has walked past the lady with the funny hat, when they sing, "If being mean's a way of life you practice and rehearse," you see the same Muppet with the same hat, now standing in front of Scrooge. She had no time to get there between the two shots.