Continuity mistake: At the mall Stewie is wearing a pair of mustard color fingerless mittens, and the problem occurs while Stewie is sitting on Santa's lap. When Stewie first mentions plutonium and being a good boy, Stewie's hands show all of his fingers which means he's mittenless, but the color of his hands is the same color as his mittens. Then, when Santa tells Stewie to wrap it up, Stewie is wearing his mustard color mittens, then his mustard color fingers return again.
Continuity mistake: While Lois is driving her family home from the mall, the animation process did not properly color Lois during the car ride. First, she's wearing white gloves, but in one shot both of her hands are bare on the steering wheel. Second, Lois is wearing a green coat and white scarf around her neck, but in the shot facing the town's tree through the windshield, her scarf/collar is red (her Santa hat has a red narrow tip with a white pom-pom).
Continuity mistake: When Lois reminds Peter to go get the tree, there's a wreath with a bow hanging on the wall between the two pictures behind the couch, but when Lois convinces Peter with the promise of a new VCR that wreath has vanished, then it reappears and disappears again when Peter returns with the tree. Also, the plate of cookies that are on the arm of the couch seem to vanish as well.
Continuity mistake: When Lois asks Brian to put on his sweater the rabbit ears antenna is atop the TV, but when Lois asks Peter to "please" get the tree, the TV antenna has vanished then it reappears, and the rug isn't where it should be then it too reappears properly.
Continuity mistake: When Brian is in the kitchen with the oven fire and an empty fire extinguisher, the side door behind Brian has its door handle on the right side, but when the family returns and Lois opens that door, the handle is on the left side of the door.
Continuity mistake: When Peter is driving, Quagmire is sitting beside him and Cleveland and Joe are in the back seat. Joe has brown hair and he's wearing a gray knit hat and a light coat with a gray fur collar. When Quagmire is sticking his body out of the car door and yelling "Whooooooa! Whoooooooa!", look at "Joe" in the back seat. Joe doesn't look like Joe in this shot. He's wearing a Santa hat and a brown coat just like Quagmire's, and a bit of his hair is also black like Quagmire's, instead of brown. (00:05:30)
Audio problem: After Lois goes insane and jumps out the window, she insults Frosty the Snowman, who then goes up to her to demand an apology. Lois then takes out a match and uses Frosty's nose to light it. Frosty recoils, holding his nose and saying "Ow, ow, what the-" but it cuts to a shot of Lois grabbing a bottle of beer from a hobo as Frosty is saying "What the" and it shows him standing perfectly straight, with his mouth not moving at all.