Crew or equipment visible: In the episode where Beaver and Gilbert work as golf caddies, the shadow of the camera and a crew person can be seen on Gilbert as he leaves the scene.
Revealing mistakes: "Beaver Gets a Haircut" episode (Episode 4): When Wally and Beaver are talking in their room, right before Wally calls Beaver a "goof," you can catch the Beaver mouthing Wally's line along with him.
Continuity: In "The Shave", June is holding a plate with Beaver's dessert on it. In the very next frame, she is still holding the plate but the dessert is gone.
Miscellaneous: During the opening credits of the '62-'63 edition of the show, the main characters can be seen coming out the front door of the house, and one can see the inside. There appears to be some sort of entryway, with a table with flowers on it. Subsequently, in interior shots, there's nothing like that to be seen.