Continuity mistake: When Homer and Barney are talking about the lottery, Barney says, "130 million." Then, they set their mugs on the table but when they all run off their mugs have disappeared. (00:01:50)
Continuity mistake: The colour of the sign Moe switches on at the start changes from yellow to red. (00:01:50)
Continuity mistake: Homer asks Bart and Lisa what they will do with their money, Bart has a can of Buzz cola in his hand and when Lisa speaks the can simply vanishes from Barts hand. (00:04:20)
Continuity mistake: When Homer is covered in gold and goes into Burns' office, Burns has 2 folders on his desk. One yellow and one white. However, the white folder becomes yellow in one shot. (00:04:50)
Continuity mistake: When watching the draw, the family have their TV dinners on their lap, which seem to instantly disappear between shots. (00:05:10)
Continuity mistake: When Bart is in the car and asks when the dog is going to get his operation, his hand is not on Lisa's seat. When the shot changes, to show the whole car, it is. (00:07:50)
Continuity mistake: When Marge is listing all the cut backs, the calculator on the table keeps appearing and disappearing between shots. (00:10:30)
Continuity mistake: When Homer suggests the family becomes a group of acrobats, a clipboard suddenly appears on the table in front of Marge. (00:11:30)
Continuity mistake: When Bart is pacing waiting for Sant's Little Helper, the time on the clock changes drastically when the nurse opens the door. (00:11:40)
Continuity mistake: When Santa's Little Helper is eating from his bowl after his operation, the amount of food in his bowl increases drastically. (00:12:50)
Revealing mistake: When Homer is putting up all the lost dog posters on the building that gets blown up, the stapler isn't on the poster. (00:17:30)
Continuity mistake: When Homer tells Marge to go and get all her money, he is holding her shoulder next to her necklace. Camera changes and his hand is lower.
Audio problem: When Bart says "I'm going to find my dog" his mouth does not match what he is saying.
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