Continuity mistake: At the beginning of the show, two men walk into the courthouse. (As a sidenote, one of the men is Rance Howard, which is Ronnie Howard's real dad). Barney is the only one there, as Andy is out. The two men sit down to wait for Andy. Rance Howard is not wearing a hat, and he is in a light-colored suit. When Opie walks through the door looking for Andy, you can see that Rance is not sitting there any more, and now it's a man with a dark suit and wearing a hat.
Factual error: Goober and the Taylors stop and get gas in Raleigh before heading back to Mayberry. As they are waiting, a Coors Brewing Company truck is seen driving past the service station. Coors Beer was not sold east of the Mississippi River until the mid-1980s.
Revealing mistake: When the bus pulls up for the Taylor family to leave for Hollywood you can see behind the bus a California freeway with cars moving in both directions. I don't think that Mayberry had a freeway right outside town. The scene is fixed when they return in "Aunt Bee on TV." There has been a tree put there so the road wouldn't show.
Continuity mistake: In establishing scenes throughout the first part of the episode, Andy is seen driving up to Meyers' home which seems to be located far from neighbors and on a dirt road. In the "renovation" scene and those following, Frank's home is located on a paved road with other homes nearby in the shot. Clearly a different set entirely.
Other mistake: Aunt Bee refers to losing the pickle contest 10 years in a row (which Clara corrects to 11, leading us to believe they have been in the same competition), but this is her 2nd year in Mayberry, having not seen Andy for years in her season 1 intro as the "new housekeeper", (or Opie - ever) in season 1.
Factual error: Halfway through the show, Goober takes the motor out of the car and they start it up in the Sheriff's office. If you look at the direction of the fan, it would be blowing the air forward instead of towards the rear of the engine. It would be fighting the air coming through the radiator instead of pulling it.
Continuity mistake: In the scene where the meeting participants are in the kitchen, the Russian guy opens the icebox, does not close it, and sits at the table. When Aunt Bee comes in immediately thereafter, the door is closed. She opens it again and partially closes it; but in the next shot, it is fully open.
Continuity mistake: About 2/5 through the show, Barney comes into the office with a family tree chart. The edge of the paper is on the bottom of the roll sticking out to the left, there's a cut to a different angle and the edge of the paper is on top, not sticking out and still facing to the left. On top, it should be facing to the right.