Continuity mistake: When Bart arrives home after his bad day, he is seen wearing his lucky red hat. It was never visible anywhere before he got home. (00:07:10)
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.
Audio problem: When Grandpa Simpson is typewriting a letter in his room, the words he's spelling out loud don't match his fingers hitting the keys.
The Simpsons Halloween Special: Tree House of Horror - S2-E3
Continuity mistake: After the aliens have dropped off the Simpsons in their back yard, Marge is seen wearing her usual red pearls. Except in the final shot of the family. Then her pearls are white. This is not due to the alien light source altering the colors in the segment, because Marge's pearls have been a dark color through the whole episode, even inside the alien spaceship.
Skinner's Sense of Snow - S12-E8
Continuity mistake: When the children are sleeping in the canteen at school, Bart is resting his head on a red sweater. In the close up of him and Lisa, the sweater has now turned dark green. It's not because of the dark that we see it like that because there is a bit of light at the beginning of the next shot and we see the jumper to be dark green.
Continuity mistake: The "Stinger" only appears on the boxing gloves after Moe passes them to Homer. It is not there in all preceding shots.
Factual error: When Bart declares war on Nelson, Herman uses the declaration from the Franco-Prussian war and puts "Bart" in place of "Otto Von Bismarck". The thing is, Bismarck didn't declare the Franco-Prussian War, Napoleon III of France did. A small error, maybe, but something that a war buff like Herman should know.
Sunday, Cruddy Sunday - S10-E12
Continuity mistake: When Homer tells Lisa who is going with him to the Super Bowl, he says Comic Book Guy is going. In the shot showing the people lining up, Comic Book Guy is not there. He then appears on the bus later on.
Tree House of Horror X - S11-E4
Continuity mistake: When The Simpsons go home they find the words 'I know what You Did' on their door, written in blood. Then another shot after, you can see the door with nothing on it.
Scuse Me While I Miss the Sky - S14-E16
Other mistake: Bart and Lisa overload the nuclear power plant to create a city-wide blackout so everyone can see the meteor shower. All the lights in Springfield blow out. Yet when we see the Simpsons sitting on their lawn watching the shower, every single window in their house is yellow, as if every room has their lights on. It is the only house shown with power.
Continuity mistake: After Homer pushes in front of Lisa in the queue for the bathroom, she loses the toothbrush she was holding in her left hand. (00:01:15)
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)
Tree House of Horror III: The Simpson's Halloween Special III - S4-E5
Continuity mistake: When Homer is escorting his family out of the house to escape the Zombies, he leaves the door wide open, but in the very next shot it is shut. (00:18:50)
Springfield Connection - S6-E23
Continuity mistake: When Marge is reading the books on death sports the police arrive and she turns around, you can see the two posts with the round balls on top with no words on it. But when Marge starts walking towards the police station it says "Police" on the balls and the post changes size. Also the door to the police station changes from brown to blue to clear glass and back to blue during the scene.
Continuity mistake: While Bart starts the I.Q. quiz and is reading it out loud, his teacher says "Shh. Visualize it Bart." While this happens if you look in the background you can see the alphabet line. It starts out Tt Uu Vv Ww, then Bart's teacher blocks out some, then you see the letter T again. (00:04:50)
Life on the Fast Lane (a.k.a. Jacques to Be Wild) - S1-E9
Continuity mistake: At work, the salad in Homer's sandwich disappears when he mentions that 'she' made it.
Continuity mistake: When Bart says "Everybody, come quick!", the fridge door goes from open to closed.
The Last Temptation of Homer - S5-E9
Continuity mistake: When Bart has had his throat sprayed, we see him standing up just before the shot changes to the power plant. The shine on his shoes instantly disappears and reappears several times.
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.
Continuity mistake: After Homer dodges the cannonball near the end of the episode, the blue platform that he stood on goes further away from the edge of the stage between shots, and when Homer hugs Marge the platform has disappeared and Peter Frampton's cooler has become closer to Homer.
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