The Simpsons Christmas Special: Simpsons Roasting on an Open Fire - S1-E1
Continuity mistake: Lewis changes from black to yellow when Milhouse dares Bart to pull Santa's beard off. (00:14:10)
The Simpsons Christmas Special: Simpsons Roasting on an Open Fire - S1-E1
Revealing mistake: After Homer brings Santa's Little Helper home, Homer and Marge hug, and if you look behind them, the background is upside down. There is a picture hanging upside down, and there is a lamp which seems to be hanging from the ceiling. Then in the next shot, the background is normal. (00:21:40)
The Simpsons Christmas Special: Simpsons Roasting on an Open Fire - S1-E1
Continuity mistake: While having breakfast, all of the Simpsons' glasses of orange juice keep disappearing and reappearing throughout the scene. (00:05:35)
The Simpsons Christmas Special: Simpsons Roasting on an Open Fire - S1-E1
Continuity mistake: When Homer walks out Circus of Values with his purchases, he bumps into Flanders, and all the presents scatter everywhere. Watch out for a red present with a green bow that goes flying near the left hand side of the screen. As soon as it hits the ground it turns into a pink box with a white bow wrapped around it, with no shot changes. (00:10:15)
The Simpsons Christmas Special: Simpsons Roasting on an Open Fire - S1-E1
Continuity mistake: Bart yanks Homer's beard off in the mall, Homer then takes Bart into Santa's workshop. When they are inside Santa's workshop, Homer's beard is back on, despite the fact that he let it fall to the ground when Bart pulled it off.
Moaning Lisa - S1-E6
Continuity mistake: Marge drops Lisa off at school for band practice. She walks up to the entrance steps, and then converses with two boys. Then her teacher comes up, and Marge gets annoyed. When she comes to grab Lisa back into the car, Lisa is right by the roadside, along with the two boys and the music teacher.
The Simpsons Christmas Special: Simpsons Roasting on an Open Fire - S1-E1
Revealing mistake: When Marge is speaking on the phone to her sisters, her sisters say something that sounds like they said that Homer is ugly. Marge then puts down the phone wire that she was curling round her finger. You can temporarily see that the phone wire disappears when Marge puts it down. It is at the very end of an early scene, because as soon as this happens it changes to another scene. (00:04:40)
Factual error: After Bart's prank call with Moe, you can see Bart and Lisa laughing. However the telephone's wire does not connect to the receiver. (00:09:40)
The Simpsons Christmas Special: Simpsons Roasting on an Open Fire - S1-E1
Continuity mistake: When Mr. Burns comes on the P.A. system Homer started eating a donut, he takes a bite and after that the donut stays the same size even though he takes 2 more bites. (00:06:40)
The Simpsons Christmas Special: Simpsons Roasting on an Open Fire - S1-E1
Continuity mistake: Marge walks through the doorway from a room with a piano, beside a bare wall, and on the piano are two frames and music sheets. But when Marge turns the jar upside down, there is a large picture on the wall beside the piano, and the music sheets and frames have disappeared from the piano. (00:08:00)
Audio problem: Just after Mr Smithers says "...so that we have the energy to from your favourite video game to yummy cotton candy machines", all the children start clapping but the sound of clapping starts BEFORE they start clapping. (00:05:05)
Revealing mistake: During the first ten minutes when the twins are explaining to Bart what incompetence means, one of the twins' body disappears, showing a floating head and hair. (00:06:30)
Audio problem: Both Lenny and Carl's voices are wrong. At one point Carl's voice is higher pitched, and Lenny keeps speaking like Moe, or like a voice we haven't heard before.
The Simpsons Christmas Special: Simpsons Roasting on an Open Fire - S1-E1
Continuity mistake: When Homer finds out that he has no Christmas bonus the background behind Marge is the kitchen, but then it cuts to Homer and the background behind him is the kitchen too. (00:08:20)
Continuity mistake: When the cops pin Krusty to the carpet, he moves further away from the yellow stripes between shots.
Continuity mistake: When Bart's army is attacking Nelson, there are a lot more kids attacking Nelson than there were training with Bart.
Continuity mistake: When Sideshow Bob is shouting and screaming just before he is taken away in a police van, Lisa and Bart swap places right before Bart says "Take him away, boys."
There's No Disgrace Like Home - S1-E4
Other mistake: When Homer is at Moe's, there is a boxing match happening on TV. There is a bald guy who is getting considerably battered by the opponent and gets knocked out, and then for some absurd reason he is victoriously standing and smiling as if he had won.
The Simpsons Christmas Special: Simpsons Roasting on an Open Fire - S1-E1
Continuity mistake: During the family's breakfast, in most shots Marge sits right beside Maggie's highchair, but in certain close-ups they are quite a few feet away from one another. (00:05:25)
Moaning Lisa - S1-E6
Continuity mistake: When Homer says "I can't concentrate with that infernal racket", the shot goes up onto Lisa's room. From the outside, there is no window frame on the window. But from the inside, there is one going right through the middle.
Answer: I'm not sure if this is the one you're thinking of, but an episode of "Ray Bradbury Theatre," called "A Sound of Thunder," dealt with a similar matter: a group of hunters travel back in time to hunt dinosaurs, only to find things have changed when they get back because someone stepped on a butterfly.
Xofer