Continuity mistake: In the garage, Martin walks behind Lewis and the other kid. He jumps to walking in front of them in the next shot. (00:09:05)
Yoshi
Home Sweet Home-Dum-Diddly Doodly - S7-E3
Continuity mistake: At the Flanders' house, in the kitchen, Bart and Lisa have sandwiches on plates in front of them. The shot changes and everyone on the table has an empty bowl including Bart and Lisa.
David Mercier
$pringfield (or How I Learned to Stop Worrying and Love Legalized Gambling) - S5-E10
Continuity mistake: As Bart is on the phone trying to book his new act, you can see there is a split down his white shirt. When the shot changes and he tells Lisa to be quiet you can see that split has disappeared. (00:15:05)
Lummie
Continuity mistake: After Bart cleans his hands with a cloth after going to the private toilet, he puts it on his shoulder. A couple of shots later it has disappeared.
OL1V3R666Continuity mistake: When Selma is on video at the end of the episode, the iguanas's eyes change from yellow to white during the scene, but not between shots.
OL1V3R666Continuity mistake: At the end of the episode, Bart is writing lines, and he has already written on one half of the board. On the half he has already written lines on, the lines go down the board about 3/4 of the way, yet when the shot changes, the lines now go all the way down to the bottom of the board. (00:20:35)
Hamster
Alone Again Natura-Diddly - S11-E14
Continuity mistake: When Homer is filming Ned cutting the bushes between the two front yards, the camera Homer is holding suddenly switches eyes.
Continuity mistake: When Bart is writing from the books in the kitchen, he is seen writing on the paper. In the next shot of him, the paper is clean.
Continuity mistake: In the scene at the beginning when the Simpsons are eating dinner, Marge and Homer's plates are empty, then they have food on them, then they are empty again. (00:03:40 - 00:04:30)
YoshiLisa vs. Malibu Stacy - S5-E14
Continuity mistake: When Lisa and Marge go on the tour, when the film has finished the woman turns it off with a remote controller and puts it behind her back. In the next shot from behind her it isn't in her hands and there is no sign of a pocket or any indication she put it anywhere else.
David Mercier
Continuity mistake: When Homer apologizes to Moe at the end, he comes in while the door is left open. It remains closed for the rest of the scene. (00:21:05)
YoshiBrother From Another Series - S8-E16
Continuity mistake: When in the dam Bob stops the motors from spinning. Next shot shows the blades spinning again, but when the camera cuts they are stopped again.
Ssiscool
Continuity mistake: In the scene where Moe is cleaning his window inside the bar, take notice that the size of the spot he cleans on the inside of the bar is small. But when the shot changes to the outside, with Moe and the others looking out from the inside, the clean spot is larger.
RLNHello Gutter, Hello Fadder - S11-E6
Continuity mistake: When Homer is talking to Ron Howard on the street, there is no pocket on Homer's shirt. But when he takes the money from Ron, he puts the money in a pocket that suddenly appears.

Moaning Lisa - S1-E6
Continuity mistake: When Marge says, "Oh Homer, you'll lose your head if it wasn't secured to your neck," her necklace is white. However, when she gives the cup cakes to Homer and Bart, it is red. (00:02:00)

Continuity mistake: After the dinner with Sideshow Bob, Patty, Selma and the Simpsons, the food on Bart's plate changes between shots. We first see a piece of meat with some peas, then we see two pieces of meat and some mashed potato. It changes randomly throughout the scene. (00:02:40 - 00:07:05)
OL1V3R666Continuity mistake: After Krusty is done doing the special promotion commercial, he is given a telegram informing him of the Soviet boycott. However, this is a flashback, and at this point in time Krusty is illiterate (as will later be established in season 1) so he shouldn't be able to read the telegram.
Continuity mistake: When Lisa enters the taxi and we see the door being closed, the camera turns into the full view of the taxi in the next shot, and now we see that the door (the one with company logo on it) has changed its shape between shots.
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