In every episode in season one where the exterior doors are seen, the locking mechanisms have no exterior keyhole or interior locking knobs. This can be seen here when Ward is saying goodbye to the Beaver at the exterior front door, and when June opens the exterior kitchen door in one of the last scenes.
When Ward and June are checking to see if Larry is okay in bed at midnight, Ward makes the comment, "Maybe we should call the doctor." This is back in the day when doctors actually made house calls for children. But even then NO doctor is going to make a house call at midnight.
In every episode in season 1, where you see the family car pulling into the driveway, the driveways is always wet, and there is a puddle at the end of the driveway on the road, yet the rest of the road is always dry.
Listen closely as Wally complains about Larry Mondello staying overnight, June mispronounces his name! She says, "Wally, what's wrong with Larry Mondella?" instead of Mondello.
As Ward is tending the hamburgers over the grill it's easy to tell that the grill isn't actually lit.
When Beaver is in the hallway outside Ward's bedroom and goes to switch on the light, the light comes on early - before the sound of the light switch and the visual of Beaver flipping the switch.
When Ward is on the phone calling a cab, Larry and the Beaver look at each other angrily. Then Larry prematurely sticks his tongue out at the Beaver and looks over at the camera with a look on his face that shows he knows he was too fast to do this in the script. Then after Ward is off the phone,, the camera goes a close-up of both Larry and the Beaver sticking their tongues out at each other as the script probably wanted.