Continuity mistake: As Martin reveals his Honor Rollor at the time trials, we can see Bart on the right side of the screen. He suddenly appears beside his car in the next shot. (00:13:00)
YoshiContinuity mistake: When Nelson is handed the knife by his two friends, Martin, Bart and their cars have disappeared for two shots. (00:13:45)
YoshiContinuity mistake: As Nelson crosses the finish line at the time trials, The Front piece of wood on Bart's car breaks off. It is attached to the car again in the next shot. (00:14:10)
YoshiContinuity mistake: When Homer sulks on the couch, he has a pillow under his head, which is pink. In a closer shot, the pillow turns light blue, but in another shot, the cushion has turned pink again.
Continuity mistake: When Homer laughs at Patty and Selma, for a few frames you can see that the beer can he was holding is gone. It stays beside the phone for the rest of the scene. (00:07:00)
YoshiContinuity mistake: When Homer helps Bart with his racer, the wall of the inside of the garage keeps changing from blue to red. (00:11:45)
YoshiContinuity mistake: When Homer jumps in the car having completed the Fatherhood test and drives off to see Bart race. The car shifts from reverse to forward without Homer moving a gearstick. (00:18:40)
Continuity mistake: When Homer calls the Fatherhood Institute about his poor test results, the guy is holding a cup with 'Super Dad' on it. In the next three shots the cup has jumped to being on his desk. (00:08:50)
YoshiContinuity mistake: Before the time trials start, the audience on the sidelines keep disappearing and reappearing. (00:12:30 - 00:13:40)
YoshiContinuity mistake: At the institute, Dave puts his hand on his son twice, and they also swap positions between shots. (00:10:00)
YoshiContinuity mistake: At the Institute, the clipboard Dave is holding repeatedly switches arms between shots. It also disappears entirely. (00:09:35 - 00:10:35)
YoshiContinuity mistake: Patty sprays Barney with pepper spray. It disappears from her hand in the next shot. (00:18:55)
YoshiContinuity mistake: At the end, when Bart gets his trophy, Marge and Homer switch positions. (00:20:35)
YoshiContinuity mistake: In the video store, Homer grabs the 'Football's Greatest Injuries' tape with his right hand. He was also holding Maggie with his right arm, so he should have dropped her. (00:06:00)
YoshiContinuity mistake: In the video store, right after Homer picks up the tape, Maggie switches arms. (00:06:05)
Yoshi
Continuity mistake: Nelson makes a hole in the Honor Roller during the finals. This hole disappears and reappears between shots. (00:20:00)
YoshiContinuity mistake: When Dave gives Homer the Bill Cosby book, the cover is light blue. When he reads the book in the garage, it is green. (00:09:40 - 00:11:30)
YoshiContinuity 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)
YoshiContinuity mistake: In the video store, when Homer looks at sports tapes, the categories of movies behind him change from drama, horror, and 'Elvis' to drama, western and horror. (00:05:55)
Yoshi
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