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: 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.
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.
Plot hole: About 2/3 through the show Barney is locked inside the armored car and Gomer puts about a gallon of gas into the gun port onto Barney's shoes. After Gomer is told of his error, Barney stays inside the truck. The fumes would have killed him, not to mention the risk of explosion.