Revealing mistake: In all the intro scenes from season 2 to season 20, episode 9, 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: During the intro to some earlier episodes where Bart leaves the school, the scene pans down and the edge of the animation paper can be seen in a couple spots at the very bottom of the frame. It is blue in color. This error persisted through many seasons.
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)
Revealing mistake: While Patty tells Selma "Look who's talking" after Homer debuts his new haircut, both of their chests up to their necklaces are the same color as their dresses. Selma's dress also suddenly has sleeves on it.
Revealing mistake: When talking to the doctor about using his health insurance, Homer's hairline above his ear vanishes in a shot then reappears.
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.
One Fish, Two Fish, Blowfish, Blue Fish - S2-E11
Revealing mistake: In the credits it shows Homer eating pork rinds. This is a reused scene from "Moaning Lisa."
Revealing mistake: While Homer is waiting for Marge to come downstairs, both Patty and Selma are wearing turtlenecks. When Homer thanks Mr. Bouvier, in the next wide shot as the twins nod their heads, both of their turtlenecks are not colored in properly at their torsos, so the background is visible through their bodies. (00:16:30)
Homer vs. Lisa and the 8th Commandment - S2-E13
Revealing mistake: When Homer is imagining being locked in the jail cell and he turns around to find he is in his own home, he turns to look at his family. There is a close up shot of his terrified face, and the people in the background watching the game (I think it might be Lenny and Carl) have no eyes. The rest of their face is fine, but their eyes are missing.
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.
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