Visible crew/equipment: When Aunt Bee drives her car up to the gas pump at Wally's station and then wins the five dollars, the reflections of crew and filming equipment are visible on the surface of her car. Also, note that the chrome trim around the windshield is covered with black tape, presumably to prevent reflection. Additionally, during all the scenes at Wally's station, the back end of one of the quonset huts from the "Gomer Pyle, USMC" set are visible, either at the left side of the screen or in the background.
The Statue - S7-E23
Audio problem: When Aunt Bee is introduced at the start of the ceremony, in the first wideshot as she begins her speech we hear her say, "Friends, neighbors, and fellow citizens of Mayberry" but it's not in sync with her mouth movements. (00:18:45)
The Statue - S7-E23
Other mistake: At the end of the ceremony, when Aunt Bee finally unveils the life-size statue of Seth Taylor the prop jiggles (presumably it's plastic) back and forth on the base, even though we know it's carved out of a block of granite (stated by Howard earlier in the episode) by a stonemason. (00:05:00 - 00:22:55)
The Statue - S7-E23
Continuity mistake: At the end, when Aunt Bee concludes the reading of the minutes and mentions the improvement fund, she's holding the blue folder up in her hands with its notepapers inside, but in the next shot one of the notepapers is suddenly flipped over the top of the folder. (00:23:45)
Aunt Bee's Restaurant - S7-E21
Continuity mistake: While Jack is trying to help Aunt Bee understand her fear, in a few of the closeups of Aunt Bee and Andy (seated on the couch) note the fruit bowl has moved and the candlesticks have vanished from the server, the large figurine on the left side of the hutch has vanished, and the flowers from the table have also disappeared. Then everything is back in their place. (00:21:10)
Aunt Bee's Restaurant - S7-E21
Visible crew/equipment: After Goober orders the chow mein dinner at the restaurant, in the following wideshot of everyone at the table, the curved chalk actor's mark is visible on the floor between Andy's and Helen's chairs. (00:12:20)
Andy's Old Girlfriend - S7-E20
Other mistake: At the cabin, when Andy and Alice are fixing breakfast at the kitchen counter, note the cereal box's brand name is covered (presumably painted), which is still recognizable as "Post Toasties." (Andy Griffith even did commercials for this specific cereal). (00:20:00)
A Visit to Barney Fife - S7-E18
Other mistake: After Andy and Barney hear on the radio that another supermarket has been robbed they head over there, and in the shots at the supermarket note that many of the products on the shelves have their brand names covered with tape or scribbled over. For example, boxes of "Mrs. Cubbison's" stuffing/dressing and "Sunshine Cheez-It" are covered with tape. (00:16:40)
A Visit to Barney Fife - S7-E18
Visible crew/equipment: When Barney introduces Andy to Ma's boys, the actors' two tape marks are visible on the floor in front of the couch. (00:07:10)
Don't Miss a Good Bet - S7-E16
Visible crew/equipment: When Andy walks into the diner to speak with George Jones about investing in the treasure hunt, the shadow of the overhead boom arm and mic is moving around on the wall behind them. (00:18:30)
Other mistake: Twice during the episode there's a brick wall adjacent to the grocery store which should not be there. First when Clara walks down the street toward Floyd, then later when Andy and Opie also walk down the street toward Floyd. (Shot in the studio not backlot). (00:02:05 - 00:16:55)
Continuity mistake: When Helen and Howard are talking about the lack of funds for the school play Andy comes up with a solution, and as Helen walks up the street there's a brick wall adjacent to the grocery store window which shouldn't be there (shot on the studio soundstage not the backlot). (00:07:05)
Continuity mistake: When Aunt Bee and Opie drive up to Wally's station to get gas, the car's vent windows are either open or closed depending on the angle of the shot. (00:06:05)
Visible crew/equipment: When all the Darlings walk out of Andy's kitchen with a bag full of Aunt Bee's muffins, the actor's tape mark is visible on the floor, where one of the Darling boys looks down and stands. (00:20:25)
Continuity mistake: During the baseball game, when Opie slides into homeplate we can see the concrete mixer truck in the background beside the building, but when Helen shows Aunt Bee the photo she took of Opie the truck is gone. (00:09:40 - 00:23:25)
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.