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)
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
Revealing mistake: When Homer brings Santa's Little Helper home, a single frame of Maggie falling off Marge's lap is misplaced.
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)
There's No Disgrace Like Home - S1-E4
Revealing mistake: When the Simpsons are eating at the dinner table, Marge's necklace keeps flashing white.
The Call of the Simpsons - S1-E7
Revealing mistake: The scientist who describes Homer as a brilliant beast seems to have some of his hair magically floating above his ear - it's not connected to his head, yet it moves alongside his head. (00:20:40)
Revealing mistake: When Barney says "How long will the story take?" after Bart offers to tell the mob his story, the upper half of Barney's face starting from his nose is a darker shade of yellow than his lower half. His shirt also changes colour. (00:01:40)
Revealing mistake: In the beginning, when the angry mob is chasing after Homer and Bart, there are two Dr Marvin Monroes. One of them is behind Mrs Krabappel and the other is next to Grampa Simpson. (00:01:40)
Revealing mistake: The blonde woman's midriff (from the Space Mutants movie) flashes pink for a frame.
Revealing mistake: When Bart makes a copy of his photo for Lewis, footage of Milhouse inserting a coin into the copy machine is used. You can see Milhouse's pink shirt sleeve instead of Bart's orange shirt sleeve. This same shot is again reused a third time seconds later. (00:09:55)
Revealing mistake: When Homer enters the Kwik-E-Mart, the automatic door doesn't fully open, and Homer's left arm goes right through it. (00:03:40)
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