When Fester first comes out of the escape chute, he comes out of Pugsley's, which is the left one. Later, when Wednesday escapes before the Mamushka, the escape chute covers are switched. When Tully, the doctor and Fester are trying to get into the vault and come out of the chute, the covers are switched back.
During the school play, Wednesday's cheek is smeared with blood at her death scene. When she turns to face the audience, spraying the front row, the blood on her cheek is no longer there.
When Morticia goes back to the Addams mansion to speak to Tully and is captured, and Thing follows her and goes back to the motel to tell Gomez, it's obviously night. But when Gomez is at the table and Thing is trying to communicate, the room is filled with daylight. When they leave to save Morticia, it's night again.
When Wednesday and Pugsley are selling lemonade there is only one sign behind them, but when Morticia goes to talk to Fester, she walks by the same spot and the sign is moved and another is there with the original next to it.
When Margaret is examining the Chinese finger-trap with Morticia, she inserts both fingers in the device, which would trap them. In the next shot, her fingers have escaped the trap. Then she inserts them again and they become stuck.
The Addams' lose their house when the judge assigned to handle the case turns out to be their next-door neighbor who despises the Addams family and happily sides with Fester, causing the Addams' to be kicked out. In reality, this would immediately make the case into a mistrial and the verdict would be dismissed on appeal since the judge clearly allowed a personal grudge against one of the parties involved to prevent him from making an unbiased judgment.
A judge would never be allowed to preside over a case where their neighbor is involved for the reason exactly shown in the film; there is a risk of the judge's personal feelings towards their neighbor preventing them from making an unbiased assessment.
Lurch burns the wooden Indian outside the motel by breathing fire on to it. In later scenes, the Indian is restored.
When Thing is Morse-signaling Gomez after Morticia has been kidnapped, tapping with a spoon on a desk, a part of the table can be seen wobbling, revealing a gap for the actor to put his hand through.
Obvious use of knives juggling doubles for Gomez and Fester in the Mamushka sequence.
When Thing is trying to tell Gomez about Morticia, he uses what appears to be finger spelling from American Sign Language. However, he doesn't actually spell any words, but signs random letters, mostly T, M, N, R, and a few others, and several other random hand positions that aren't letters at all.
When Thing is pulling the wagon with his belongings, the string pulling the wagon is visible.
When Gomez tells Tully he cannot discuss new business until the next quarter, he flips a desk calendar to that day's date, which is Monday, September 9, 1991. The opposite page of the calendar says Thursday, August 8, 1991.
Gomez puts on some 16mm home movie reels from when he and Fester were children, from the 1930's. This film takes place in the 1990's, which would place both Gomez and Fester at about sixty years old. Elsewhere in the film, Fester's date of birth is listed as 1947, and Gomez is younger than him.
When Fester climbs in his bedroom window after chasing Wednesday into the cemetery, you can see the harness on the back of his collar.
When Wednesday tells Dr. Pinderschloss "Nobody gets out of the Bermuda Triangle, not even for a vacation" she knows that her own parents spent their honeymoon in the Bermuda Triangle and, therefore, must have got out.
The Addams family motto in Latin "Sic gorgiamus allos subjectatos nunc," actually translates to "So let's enjoy the wings now". A closer quote would be: "qui nos opprimere velint, illos libenter devoramus" which translates to "Those who wish to oppress us, let us gladly devour them".
Fester tells Wednesday to "Aim for a major artery: the jugular!" However, the jugular is a vein; the major artery in the neck is the carotid.