Beaver gets up from the breakfast table with his napkin tucked in his pants as he moves away, but in the next shot, the napkin is on the table.
When the boys jumped down from the tree, Gilbert, Whitey and Larry were in that order, next to Ms. Landers; however, the next scene has Larry, and Gilbert on the opposite sides.
Gilbert is not seen in Beaver's classroom even though he is supposed to be in Beaver's class as one of Miss Lander's students. He is only seen later in connection with Larry and Whitey and the tree.
June does something seen frequently in shows and movies, but makes no sense in real life. She calls Beaver's school, they answer [unheard by us], then she asks, "Grant Avenue School?" The school would've answered with "Grant Avenue School," and perhaps "May I help you?" or "How may I direct your call?" Only a residential phone would be answered with "Hello," which would then make the caller ask if it was their intended party.
Beaver asks his Dad not to say "Soup's on" when dinner is ready and his dad assures him he won't. When dinner is ready Ward calls out "Soup's on!".