The Taylors in Hollywood - S6-E8
Revealing mistake: While Andy, Aunt Bee, and Opie are at the Hollywood studio watching a scene being filmed, when Andy mistakenly stands up and interrupts the filming we can see the wall and shelves behind the director, and it should look familiar because it's the wall and shelves from Floyd's shop in Mayberry.
The Taylors in Hollywood - S6-E8
Visible crew/equipment: After the studio's security guard tells Andy they're filming on stage 40, when it cuts to the Taylors walking onto the set we can see two chalk T-marks on the floor (one in front of Andy's feet). (00:13:00)
Other mistake: During the scenes at Goober's filling station we can see a few brand names covered. For example, the "Chew Mail Pouch Tobacco" wall thermometer sign has its brand name "Mail Pouch" covered with black tape, and all the yellow "Pennzoil" cans have half the letters of their brand name painted yellow, so all the cans read "en zo."
The Bazaar - S6-E5
Visible crew/equipment: After Andy shouts at Goober to get his pickup truck away from the courthouse, it cuts to Andy, Warren, and all the women in the courthouse, and we can see the chalk T-mark on the floor at Warren's feet.
The Legend of Barney Fife - S6-E18
Continuity mistake: At the start of the episode, while Barney's in Andy's kitchen he's wearing a shirt with large distinctive designs, but when he and Andy walk into the courthouse Barney's shirt has completely different designs (though it's the same type of shirt). (00:01:00 - 00:04:00)
The Gypsies - S6-E23
Continuity mistake: While Andy and the townsfolk are at the courthouse during the storm, when The Gypsies walk in the two men are wearing wet rain ponchos and the woman is wearing a striped serape, but then she's wearing a wet raincoat, and it switches yet again in following shots. (00:21:45)
The Cannon - S6-E11
Visible crew/equipment: After Warren accidentally fires The Cannon, in the following closeup of Andy commenting that he shot a station wagon, the shadow of the moving boom and mic is visible at the lower left corner of the screen. (00:22:05)
Revealing mistake: When Andy arrives at the studio for the publicity shots, while he's speaking to Mr. Michaels note that behind them we can see the Taylors' Mayberry kitchen cabinet/counter and a wall with their distinctive gold/silver design wallpaper. (00:03:50)
The Taylors in Hollywood - S6-E8
Revealing mistake: When Andy, Aunt Bee, and Opie are visiting the studio set while the scene between fake Aunt Bee and fake Andy is being filmed in a fake Taylor living room, note that behind the real Taylors we can see bits of the interior of their Mayberry house scattered in the background offstage. Some things to look for include their living room's stone wall with the mantle, their two reddish chairs (one with the fringe at the bottom), and a wall with their distinctive gold/silver design wallpaper. It's particularly funny when Opie looks at the "fake" living room set and comments, "Is that our house? It sure looks different!" while we the viewers can see the "real" Taylor set behind him. (00:20:35)
Other mistake: While Andy and Helen are rearranging everything in the squad car's trunk to make it all fit, Helen picks up the large thermos and written on its bottom we see "Reggie personal." Reggie Smith was property master on set. (00:12:45)
Aunt Bee Learns to Drive - S6-E21
Continuity mistake: The car which Aunt Bee buys from Goober has a rearview mirror which repeatedly disappears and reappears between shots, throughout the episode.
Wyatt Earp Rides Again - S6-E20
Continuity mistake: After Opie tells Andy that he knows he'll win at high noon, it cuts to Warren in front of Floyd's shop (with its furniture conspicuously missing), but later when Andy walks over to Floyd's shop its facade has changed. Note differences such as window structure, and the name "Floyd's Barber Shop" painted on the window which wasn't in the earlier shot. (00:18:05 - 00:24:15)
Visible crew/equipment: After Andy agrees with Helen to suggest a blind-date for Warren, when it cuts to Warren walking into the courthouse, we can see set lighting and equipment behind the door instead of the usual painted backdrop. (00:01:55)
Opie's Job - S6-E1
Other mistake: In the scenes outside the grocery store with Mr. Doakes, there are a couple dozen cereal boxes in the display windows - Post Grape Nuts, Post Grape Nuts Flakes, Post Alpha Bits, etc., and all the boxes have part of their names covered, such as "Rap Ut." It's pretty funny considering that TAGS cast did commercials for Post Grape Nuts, and other Post foods.
Visible crew/equipment: After Goober has deliberately removed the distributor cap from Andy's squad car, when Andy slams the hood shut we can see the reflection of the filming crew on Floyd's window. Then when Andy and Helen drive away in Goober's truck, the microphone pops up into view at the bottom of the screen. (00:17:20)
Malcolm at the Crossroads - S6-E3
Continuity mistake: At the start, Andy walks down the street and stops at the church steps to watch Ernest T. being a crossing guard, but when it cuts to the shot of Ernest T. we see the church steps yet Andy's gone, then when it cuts back to Andy he's still standing by the church steps.
Continuity mistake: After Clara cancels lunch with Aunt Bee, when Bee walks over to the table there are two candle holders on the server and a large picture surrounded by four small pictures hanging over the server. However, when Opie runs to the table, the left candle holder and bottom left small picture are both gone in his closeups. (00:19:40)
Continuity mistake: During the scenes at Wally's station, the two prop gas pumps each have just one fuel dispensing hose/nozzle which switch between being between the two pumps, or with their hoses/nozzles on the opposite side of the gas pumps.
Answer: Those "boxes" are valences that used to be very common, before air conditioning. They allow for windows to be open during rain storms. They permit air circulation, without letting the rain in.