When Lurch, Gomez, and Fester are rearranging the book shelf. Some objects on the shelf that weren't there previously.
When Thing turns off the echo in the cave, and when he catches the typewriter platen, he is a left hand instead of his usual right.
After Lurch dumps the coals out, the way they are arranged changes between shots.
The Addamses are shocked to learn that everyone else in the world sees knights and heroes as the good guys in medievalist literature, and dragons and ogres as villains who must die in the story. But they already learned this and discussed it at length in The Addams Family Goes to School (1964).
As Gomez exhales what seems to be an endless amount of smoke, a stage hand can be seen at the very end ducking down after all the smoke has been 'exhaled.'
Gomez refers to Uncle Fester as having taking one lesson of "Zen Yogi". "Yoga" is the practice of the spirituality and philosophy, while a "Yogi" is one who practices it.