Deliberate mistake: The Batmobile's Bat-turn lever is located at the center upper console, between the driver and passenger seats. Whenever the Bat-turn lever is used while the Dynamic Duo are in the Batmobile, the same closeup always faces the passenger seat and shows Batman's gloved hand pushing the Bat-turn lever, but the problem is Robin vanishes in this closeup - the passenger seat is empty. Episodes S1xE3 "Fine Feathered Finks", S1xE29 "The Bookworm Turns", and S2xE38 "The Joker's Hard Times" are three examples of the Bat-turn lever being used.
Smack in the Middle (2) - S1-E2
Deliberate mistake: Ignoring that its very unlikely all of Jill St. John's copious red hair would fit under a wig that passes for Robin's hair, it's impossible that it would emerge from under that wig looking as full and perfectly styled as it does when she's in the Batcave.
Deliberate mistake: When Batman and Robin get free from the train tracks in the nick of time, the shot of the couple on the passing train is flipped; note things are backwards. (00:04:30)
The Purr-Fect Crime (1) - S1-E19
Deliberate mistake: At the end, Batman chooses the button on the right which opens the door on the right, with the tiger inside, but the last shot of the episode has been flipped, so the tiger exits from the door on the left. (00:23:25)
The Sport of Penguins (1) - S3-E4
Deliberate mistake: After the episode intro, in the first shot we see the words "The New York Public Library" above the entrance to the library, where Barbara Gordon works. However, it's stated in S3xE1 that she works at Gotham City Library, and even in this episode when Penguin listens to the news about the bomb, it's noted as the Gotham Library. (00:02:00)
The Unkindest Tut of All - S3-E6
Deliberate mistake: When Batman has figured out King Tut's plot to steal ancient Egyptian scrolls from the Gotham City Library, he and Robin rush away, and in the shot of the library exterior, the words "The New York Public Library" is quite noticeable above the entryway. (00:18:00)
Deliberate mistake: At the start of the episode the Boy Scout asks why the Cossacks are in Gotham City, then a moment later as narrator William Dozier says, "At the Gotham City Museum" the closeup we see shows "Los Angeles County Historical and Art Museum" over the museum doors.
Answer: There wasn't one singular city used. It was shot in and around various locations in California, plus on a lot of sets and backlots. Ex. Some scenes were shot in LA, some in Pasadena, some in Santa Barbara, etc.
TedStixon