Continuity mistake: When Martin does his book report, on the blackboard it says 'An afternoon with Ernest Hemingway: Reflections by Martin.' But when Bart goes up to do his report, the blackboard is clean. (00:01:25 - 00:02:20)
Continuity mistake: Bart's 1st grade drawing of cat on the fridge changes across the episode, especially its ears and whiskers. (00:04:50)
Continuity mistake: When there is a zoom in of the house just before Bart gets up late, it has the old arch-shaped windows, but in the rest of the episode it has the normal windows. It's obviously a screen from Season 1. (00:06:00)
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.
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.
Continuity mistake: When Lisa shows her exam to her family, the "a" note keeps changing from being circled in red and in a red round. (00:04:30)
Bart Gets Hit By a Car - S2-E10
Other mistake: When Marge is being questioned, before she says 'If you don't have anything nice to say, then don't say anything at all', there is a shot over her shoulder looking at the blue haired lawyer. Not only is he (the lawyer) there directly in front of her, but he is also sitting on the bench with Burns. In all other shots his chair is empty as it should be.
Character mistake: When Bart fakes being sick, on his way to the infirmary he passes the door with a sign that reads "maintanence" but the proper spelling is "maintenance", and when Bart reaches the infirmary, the sign on the door reads "ineirmary." (00:07:30)
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)
Continuity mistake: When Homer stands outside the Bouvier house to pick up Marge, the home's front door has three diagonal windows from upper left to lower right. However, when Marge gets home later after the prom, the three windows are diagonal the opposite way from upper right to lower left. (00:16:10 - 00:21:00)
Continuity mistake: When Bart gets off the bus there is a fire hydrant behind him. When we see the bus drive off the hydrant has gone. (00:07:10)
Continuity mistake: When Bart gets on the bus the day of the test, Milhouse is sitting at The Front of the bus. When we see Sherri and Terri looking at Bart, Milhouse has gone. (00:06:30)
Continuity mistake: When Homer puts Lisa's test on the fridge, a lot more papers appear on the fridge. (00:04:50)
Continuity mistake: After Mrs. Krabappel asks Bart to name the pirate in Treasure Island, when it cuts to a closeup of Bart we see the set of blue books have vanished from Mrs. Krabappel's desk, and a Krusty mug has suddenly appeared there. (00:03:00)
Continuity mistake: The door to Mrs. Krabappel's classroom has a window, but when Bart fakes being sick then walks out of the class (we see the desks as the door closes), the door's window has vanished, and the doorknob has switched sides as well. Additionally, earlier, when Bart deliberately falls off his chair in class, note that one of the bars on the back of his chair has gone missing. (00:07:35)
Continuity mistake: When Bart fails the test despite having actually studied, we see the two red "X" marks in the center row of the multiple choice questions, but after Mrs. Krabappel passes Bart because he applied practical knowledge, when Homer hangs the test up on the fridge those two "X" marks in the center have vanished.
Continuity mistake: The colour of the school changes from yellow when Martin and Bart are talking before the test to grey when Bart has spoken to Otto about being held back. (00:07:00 - 00:11:00)
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