Continuity mistake: After they say their goodbyes at the power plant, Karl asks Homer if he brought an umbrella and you can see that Homer is standing behind the doorway, but in the next shot he is now standing in the doorway.
Continuity mistake: When Homer opens the doors to the Executive Washroom, there is a sign on both doors, "Executive" on the left door and "Washroom" on the right door. When we see Homer walking with Karl and Smithers, the "Washroom" sign on the right door has gone.
Homer's Triple Bypass - S4-E11
Factual error: I get it, it's a comedy. But after open heart surgery, you are sedated for 1-2 days and on a ventilator. So, Homer even being awake after open heart surgery is completely false. Also, as someone who has had open heart surgery, you can barely move your chest because your sternum is fractured.
The Last Of The Red Hat Mamas - S17-E7
Factual error: Milhouse says he speaks fluent Italian, but his Italian is full of basic grammatical errors. He says that "insegnante" means "masculine teacher" when the same word is used for a male or female teacher. There are errors in Lisa's Italian course too ("barca piccola" instead of "piccola barca"). Despite Lisa's understandable mistakes, as she is a learner, Milhouse never corrects her (there is no such word as "idioto"). Even Milhouse's grandmother, who is Italian, makes many grammatical mistakes.
Other mistake: At the finish line, the crowd are missing all their appearance features such as faces, hair etc. (00:13:50)
The Day the Violence Died - S7-E18
Other mistake: At the end of the episode Lisa and Bart watch Itchy and Scratchy in the TV room, which is beside the kitchen at the back of their house. The TV room has a window that faces the backyard, with the record player on a shelf in front of that window. The problem is when Bart looks out the back window and sees Lester skateboarding past the house, the view outside is actually the street in front of the Simpson house, not their backyard as it should be.
Other mistake: At the concert when Bart throws the Frisbee into the dude sat next to him, the crowd in the background are all missing their eyes. (00:03:50)
Continuity mistake: When Marge spots Krusty, from the regular view, Krusty's hand is down on the table. But from the view from the camcorder, his hand is on top of the cake. (00:14:00)
Other mistake: In the scene set in the future after Lisa, as the US President, held a conference with some journalists, just before the scene changes they show a black journalist with eyes not drawn among the crowd.
Continuity mistake: When we see Marge leave Bart with Patty and Selma, they are stood with a gap between them. Camera changes to focus on the sisters, and they are now next to each other. (00:06:30)
Other mistake: When Homer pretends to read the newspaper, the back page is very noticeably blank and missing any print. (00:08:00)
Character mistake: Before their first performance, the director of "La Boheme" wishes Homer good luck. To anyone in professional theatre, doing so is the cardinal sin of theatrical superstition and is never done; the appropriate phrase is "break a leg."
Brother From Another Series - S8-E16
Continuity mistake: In the shot where Homer yells, "Marge, I'm going to look for the kids," their door is shut. After Homer sees the wave and runs back inside, their door is open all of a sudden.
22 Short Films About Springfield - S7-E21
Other mistake: When Chalmers and Skinner are having lunch with hamburgers, Chalmers bites it. When he opens it, the bread's slice is bitten but the meat is not.
Other mistake: When Moe is calling for a 'Hugh Jazz' there is a shot of the bar and Otto can be seen in the upper right corner. However his face is all distorted and out of proportion. (00:14:30)
Continuity mistake: When the robbery happens, Homer steps on the robber's feet, and we can tell he has large shoes. When they show video of the robbery on the news, the robber has regular shoes. During the hearing, they watch the video again. His shoes are back to long again.
Revealing mistake: When Nelson is offering Lisa the fraction quiz answers, right before Lisa says she's never cheated before and won't again, her right hand disappears for a single frame.
Other mistake: After Bart complains that Jonathan Harris isn't Dr. Smith, Harris says for Bart to meet him in the food court. Then the Robot says "Danger Bart Simpson, danger." How did he know Bart's name?
Marge vs. the Monorail - S4-E12
Revealing mistake: When Homer is driving along and singing, he passes three houses in the background. Those exact same houses reappear three times in the same shot even though he never turned around. (00:01:15)
Guess Who's Coming to Criticize Dinner - S11-E3
Other mistake: In the German version, the Italian cook peeved by Homer's comment on lasagna swings the cleaver at him using an Italian word ("Porcodio") which is a rather strong blasphemy, way stronger than any other expression used in original or in other localizations worldwide.
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