Factual error: Chief Wiggum visually compares a bullet taken from Homer's gun to the bullet removed from Burns and concludes that they are exactly the same. Excluding the obvious forensic violation, the bullet removed from Burns is still whole and within its casing, meaning that either Maggie took the time to remove the casing and delicately shove it in Mr Burns' bullet wound (which is dumb) or the gun fired the whole casing along with the bullet (which would have blown up the gun).
Continuity mistake: The lettering on Smithers' desk name plate changes between shots. While he is taking the call from Mr Burns' mother you can see his name "W. Smithers" in black letters. The shot after seeing Burns, who keeps falling asleep, the letters are now red.
Continuity mistake: In the 'Nightmare on Evergreen Terrace' segment, in Bart's dream you see a shadow of what looks like someone with four knife-fingers. Then when we see who it is, it is Willie holding a rake. But the rake has five tines, not four, so the shadow is wrong. (00:07:00)
Continuity mistake: When Maggie (sitting on Marge) appears on "Tyrano-Vision," Homer waves his hand in front of her. However, we can see that he isn't leaning over too far, and in the next shot it is clear that both Bart and Lisa are sitting between Homer and Marge, so there is no way he could have reached over to her in the first place.