Continuity mistake: When all the people who are plotting to murder Homer are talking about the reviews Homer had given them, Akira says, "So why did you put yours in the window?" and you see the review. At the end of that scene, when they are all laughing, you see the building they are in, and it has nothing in the window.
Other mistake: When the Simpsons' drives the electaurus off the edge of the harbour, it is shown that The Front of the car goes downwards. When it cuts to the interior view with the Simpsons' screaming, the view from the back window is tilting down, suggesting the rear is going down and The Front coming up. But in the next shot the car is seen in the water with The Front down.
Continuity mistake: The "dummy" in the episode changes consistently from a real character to a dummy sometimes. When Mel jumps into the vehicle in the museum the "dummy" moves aside. When Mel shoves it out of the way, it stays in one rigid position as if it is a dummy. During the chase, the "dummy" falls on the car behind and its facial expression changes. But when one of the guys from THAT car (not the one Mel was in) goes to the dummy, its head is ripped off and you can see that it is stuffed.