Deliberate mistake: When Marge goes offroading to get home there is a grassy field which leads right up into the Simpson driveway. However, in all other episodes there is no grassy field and instead there is a house in the spot where the field would be.
Deliberate mistake: When Bart is in the genius school, the teacher asks him if he knows what happens if he adds an acid to a base. He replies yes, adds them anyway and the lab explodes and a green goo comes out, turning Bart green. If you add an acid to a base, you just get salt and water.
Deliberate mistake: When Homer pushes the trampoline off the cliff, there is no rock below the falling trampoline. However, in the shot at the bottom a tall rock is visible which was drawn there on purpose for the trampoline to bounce off of, despite not being visible in the first shot.
Deliberate mistake: When Bart and Edna are watching "Holes" in the IMAX theater, Edna sighs and pulls out a locket which we can see the back of. After opening it up to reveal pictures of her and Seymour, we find out its actually a triptych when she opens up another image of Seymour's mother. From the way she unfolds this extra picture, Seymour's mother would have had to be facing outwards, but in the previous shot the back of the locket was blank.
Deliberate mistake: The mouth on the panda costume is in a constant smile and at no point does Homer change it, even when he is in pain. However, when he is being pulled back under the curtain the mouth is open and the mask has a frightened expression. I suppose this was done on purpose, but it's still a continuity error of sorts.
Deliberate mistake: In "Dial Z For Zombies", Homer is shooting zombies with the shotgun at the school. When he shoots the first two he just shoots them twice. Every other zombie he shoots requires him to slide the gun. It doesn't when he shoots the first couple of zombies.
Deliberate mistake: During the final musical number, just after Marge sings "I'm going bald from stress," the family sings "But we're happy just the way we are," but Homer's mouth doesn't move, because the producers forgot to record Dan Castellaneta's lines.