Revealing mistake: During the climax, one of the weasels says "Toon Town's right on the other side of the wall, boss!" If you look, you can see a brick moving completely by itself behind him. It literally just pushes out of the hole in the wall as though it's being pushed from the other side. It seems like the other weasel was supposed to be animated pulling it out, and they somehow forgot to animate that part. (He does reach over and grab it for a split second... but then immediately lets go for some reason).
Factual error: When Eddie throws Roger into the bar's storeroom (after Roger was breaking plates on his head), there's a Lucky Strike ad poster with the slogan "Be Happy, Go Lucky." However, the Be Happy, Go Lucky ad campaign didn't start until after 1948 when BDO took over advertising, and the film is set in 1947.
Continuity mistake: When Roger goes to the Acme warehouse, he brings the gold-plated gun that was in Eddie's car after Eddie took it from RK Maroon. This is a real (not-animated) gun, and when Roger is talking to Benny before going into the warehouse, he's holding a real (non-animated) gun. After Roger bursts through the drain, the gold gun he's holding is animated. (01:21:25)
Revealing mistake: Eddie heads into Toontown with an oversized animated gun. As he lurks into an alley, his gun is a plastic prop. During filming, Hoskins would carry the prop gun and the animated gun would be superimposed over it, but the animation wasn't applied to the gun in the alley. (01:13:50)
Continuity mistake: After Eddie kicks out the stool and forces a hard-boiled egg into Angelo's mouth, the eggshell fragments on the bar rearrange themselves into a neat pile. Also, two more eggs appear on the bar, which can be seen after Angelo gets on his feet and turns around. (00:13:40)