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)
Waverly Hills 9-0-2-1-D'oh - S20-E19
Revealing mistake: In all the intro scenes from season 2 to season 20, episode 19, in the wideshot where Bart skateboards out of the school, if you look closely, part of the sky can be seen on the ground. (This mistake may only apply to the original format).
Revealing mistake: When a comet is about to hit Springfield, everyone is standing on a hill in the shadow of it. But the shadow is still there even after it breaks up.
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)
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)
Revealing mistake: When everybody is in a big fight at the Do What You Feel Festival, watch out for a man in an orange shirt with black hair. His left eye keeps disappearing for one frame.
Revealing mistake: When Homer is giving Marge, Lisa and Bart money, Homer puts his hand on Bart's shoulder and whispers 'Mel's movie really stinks'. When he lets go of Bart to say this, he leaves his fingers on Bart's shoulder, but he still has fingers on his hand.
Revealing mistake: In the shot of Skinner admiring his tater tots, he is interrupted and lifts up his tray. As he inspects he tilts his tray to at least a 90 degree angle but everything, including the milk, stays in exactly the same spots.
Revealing mistake: When Smithers makes Mr. Burns his Wine Spritzer, Burns makes Homer Employee of the Month. While Smithers is pouring the stuff in the bottle, his left hand turns black for three frames. (00:10:30)
$pringfield (or How I Learned to Stop Worrying and Love Legalized Gambling) - S5-E10
Revealing mistake: At the dinner table Homer says something like "as you know it's always been my lifelong ambition to be a blackjack dealer." During this line, for some reason they appear to have used a clip from an older episode (Season 3 / Episode 3 - When Flanders Failed) and dubbed the line over. For one thing, the animation is of a much lower standard than the rest of the episode. And perhaps even more telling, when he says it, he is holding a fork with some food on it, but from the earlier shots we know there is no food in front of him.
Revealing mistake: When Homer pushes over the windmill something comes off and chops the head off a metal figure, but the head comes off before hit. (00:16:05)
Revealing mistake: When we see people digging Bobo out of the ice, they put him in a bag. If you look carefully you can see another Bobo put in another bag just before the shot changes.
Simpsoncalifragilisticexpiala(Annoyed Grunt)cious - S8-E13
Revealing mistake: When Dr Hibbert is examining Marge's hairs (and I think another time in the episode), some holes are too low and you can see through her head.
Revealing mistake: When Bart is watching Rod and Todd walk over to Ned Flanders, you can see their reflections on Bart's telescope. Their reflections do not match their actual positions.
Revealing mistake: When Barney is drinking beer at Moe's Tavern, his chest is, for one frame, the same color as his shirt. (00:11:30)
Bart Gets Hit By a Car - S2-E10
Revealing mistake: After Marge has been in court, Mr Burns' lawyer says "Thank you very much Mrs Simpson", and when he does it, the top of his head above his glasses is less pale than the rest of his face.
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)
Radio Bart - S3-E13
Revealing mistake: At Wall E. Weasel's, when Nelson is throwing balls into the 'Skeeball' machine, the animation is repeated because Bart keeps moving in and out of the camera's view. (00:04:10)
Revealing mistake: At the car convention, Homer chides Bart through the windscreen of Bonnie and Clyde's car. However, Homer's body is not seen through the holes in the windshield, which shows that the holes in the windshield were achieved via a layer of red-coloured holes. (00:04:25)
Lisa on Ice - S6-E8
Revealing mistake: After the buzzer goes on the final game and the crowd reacts one of the crowd members says "Tie Game.", then some woman in a red top says "What the Hell.". If you look at the woman when she moves her head you can see the nose on her face disappears. (00:19:55)
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