Other mistake: During the shot when Marge asks Homer who 'Hank Jones' is, you can see that the couch curves down on the left side of the screen, therefore putting her on the other side of the couch. (00:07:40)
Audio problem: When Homer sulks about not knowing anything about Bart, his first words of speech doesn't match his mouth. (00:08:20)
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)
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)
Revealing mistake: When the car arrives at the NFI, the car goes straight through the barrier. (00:09:20)
Continuity mistake: At the Institute, the clipboard Dave is holding repeatedly switches arms between shots. It also disappears entirely. (00:09:35 - 00:10:35)
Continuity 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)
Continuity mistake: At the institute, Dave puts his hand on his son twice, and they also swap positions between shots. (00:10:00)
Continuity mistake: In the garage Homer rolls up one sleeve but in the next shot both are rolled up. (00:10:35)
Continuity 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)
Other mistake: When Bart and Homer do a test drive on their race car, it is daytime, when supposed to be night-time. (00:11:50)
Continuity mistake: Before the time trials start, the audience on the sidelines keep disappearing and reappearing. (00:12:30 - 00:13:40)
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)
Other mistake: When we first see Nelson's Roadkill 2000, Nelson is moving backwards, though we see his legs as moving forward. (00:13:25)
Continuity mistake: When Nelson is handed the knife by his two friends, Martin, Bart and their cars have disappeared for two shots. (00:13:45)
Other mistake: At the finish line, the crowd are missing all their appearance features such as faces, hair etc. (00:13:50)
Continuity 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)
Continuity mistake: Lisa asks Homer to go to Dr Hibbert's office, when she does, Homer's hand on the window changes places between shots. Note the vertical bars. (00:15:50)
Continuity 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. (00:17:10)
Continuity 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)
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