Continuity mistake: When Homer tells Lisa he needs "daughter's picks" while he's watching the playoffs, Lisa tucks the pencil behind her ear with its point facing her back, but in the next shot the pencil is tucked the opposite way with its point facing her front.
Continuity mistake: After the Simpsons are selected for the 1895 reality show, they're shown the video confessional room, with the only tech in the house. When Marge is in this room confessing about her blue hair, the doorknob is on the right, but when Bart is inside confessing that he's miserable, the doorknob is on the left side of the door.
Continuity mistake: When Homer brings his family to Lenny's apartment, as Lenny shuts his front door, its doorknob is on the right, but when Homer and his family depart Lenny's crazy apartment, now that front door opens the opposite way, with its doorknob on the left.
Continuity mistake: When Marge opens the dishwasher and tells Lisa to reattach the toilet seat, there's a red dish rack on the counter over the dishwasher, but in the previous and following shots there's a filled casserole dish over the dishwasher, not a dish drying rack.
Continuity mistake: When Mr. Burns realises just how much Grayson is adored by fans, Warburton Parker has multiple bracelets on his right wrist, but in the next shot as Parker tells Burns about the media-driven world, his bracelets have vanished; then after Smithers opens the floor's trapdoor, Parker's bracelets reappear as he falls through the trapdoor.
Continuity mistake: During the episode, Grayson Mathers wears a watch on his wrist that inexplicably vanishes and reappears a few times. First, when Marge is in the school infirmary with Grayson, as he says, "I called Peyton Manning a chicken-fried Frankenstein." Second, when Grayson and Bart tussle after family dinner, and Marge tells them to behave. Third, when Mr. Burns shows up at Grayson's birthday party.
Continuity mistake: When Moe pours Pinot Noir into two wine glasses for himself and Maya, Moe's wine glass vanishes from his hand when Barney shows up. Then, when Maya acknowledges that she and Moe are definitely an item again, Maya's wine glass vanishes from her hand. Lastly, the wine bottle vanishes from Moe's right hand between shots as well.
Continuity mistake: When Moe and Maya are in the bar's back room, Moe feels like singing. At the start of this scene, there are dresses with criss-crossed straps hanging on a clothing rack, and those straps are facing the couch. As the scene progresses, changes occur: first, the criss-crossed straps face the opposite way, and then those straps disappear altogether, etc.
Continuity mistake: When Moe brings Homer to the alley outside to talk about "Midge" and his fear of losing Maya again, they go through the bar's back door, and that door's window vanishes in the following shots.
Continuity mistake: A random woman asks Homer to be her baby's godfather, and he declines. When it cuts to Moe's bar, the bar's front door opens either at its left or its right, depending on the shot. This occurs again later when Moe returns to the bar to speak to Maya.
Continuity mistake: When Ralph says, "Climate change is real," both of the cars are in the driveway, the Simpson family stands on the sidewalk, and their mailbox is gone, but after the family's close-up both cars have vanished, they're all standing on the grass, and their mailbox reappears in its proper place.
Continuity mistake: While Moe drives his car during the traffic jam, the car's rearview mirror repeatedly vanishes and reappears.
Tree House of Horror X - S11-E4
Continuity mistake: When the possible witnesses of who may have seen the Simpsons kill Flanders, Homer is shown pointing at Marge with his right hand, but the next shot shows him pointing with his left hand.
Tree House of Horror II: A Simpsons Halloween - S3-E7
Continuity mistake: When Mr. Burns is telling the robot Homer that he is his daddy, Burns is to the right of Homer, but after the POV of Burns' face (he is still oriented right of Homer) by Homer, Burns is now to the left of Homer.
Continuity mistake: When Homer ignores Lily, she gets upset then runs out of Moe's, and in the interior shot the door is hinged at the left, but in the next exterior shot that door is hinged on the opposite side, while Lily is crying outside. (00:09:35)
Continuity mistake: When Marge walks out of her aunt's house onto the porch, the screen door's handle is at the right side, but in the following shots the screen door opens the opposite way. (00:10:05)
Continuity mistake: While Lily continues to "light up all the men's lives, except Homer's," when she walks into the power plant with picnic lunches, Homer is wearing a solid peach colored tie, but in the next shot it's a striped peach and pink tie. (00:11:00)
Continuity mistake: When Homer takes Lily home after her date with Burns, as Homer pulls up to her house, the doorknob is on the left side of its front door, but when Lily enters the house, the doorknob is on the opposite side of the door. (00:16:20)
Continuity mistake: At the end, when Lily is back at a pub in England and meets Homer's doppelganger, his single shirt pocket and button are at his right side, but as he and Lily leave the pub arm in arm his shirt pocket and button are on the opposite side. (00:20:20)
Homer Alone - S3-E15
Continuity mistake: When Homer is singing to Maggie, he is standing by the bed. But in the next shot, Homer disappears and reappears.
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