The Batsignal shown in the sky did not match the silhouette on the glass - the silhouette had a visible neck, while the projected image did not. Also, the tail was longer in relation to the rest of the image on the silhouette than it was in the projected image.
Though the bat signal is used throughout season 1, it's hardly ever seen in the next two seasons, it is barely used. It nearly disappears and is barely mentioned throughout the rest of the series run, It's never explained, and is left oblivious.
In episode 7, Alfred refers to Robin as Mr. Ward, and not Mr. Grayson. But the show often refers to Dick Grayson/Robin as "Bruce Wayne's young ward." It's not a reference to the name Burt Ward.
Several times over the course of the series, Bruce refers to or addresses Mrs. Cooper as "Aunt Harriet." Strictly speaking, she's no relation to him. Obviously not, but she's Dick Grayson's aunt, so Bruce refers to her this way. Additionally "Aunt Harriet" is a warmer, more familial title than "Mrs. Cooper," so it stands to reason Bruce would use it.
Quite often when the batmobile is being rotated the black platform on which it lays can be seen; however when the batmobile leaves the batcave the platform is not there.
Since the series had a rushed production, some props from one episode found their way (sometimes erroneously) into another.
In some episodes, during fighting scenes, Robin's (Burt Ward) stunt double can be spotted.
At times during episodes, Bruce Wayne and/or Dick Grayson (and later sometimes Barbara Gordon) are able to change into (or out of) their hero costumes very quickly (or wouldn't have enough of an area to do it). But this is a --common cliche in crime-fighting hero incarnations by their artistic license.
Anytime when Batman and Robin are climbing up a wall, you can see the strings used to hold their capes up, as the scene is clearly a rotated shot of the duo walking horizontally. Although this has been widely regarded as a mistake, it's been explained (even by Adam West and Burt Ward themselves) that it was done purposely to have some fun for the adult audience.
The road sign just outside the Bat Cave notes 14 miles to Gotham City. Wayne Manor, which of course sits above the Bat Cave, is always said or depicted to be within Gotham City limits.
Gotham City is supposed to be its own fictional city. But often in rear-projected Batmobile footage, Jack Dempsey's Restaurant, a popular New York landmark of the day, can be seen.
In some episodes, Alfred is correctly recognized by villains (or allies) as Bruce Wayne's manservant. In other episodes, the same villains recognize him as a different person (when he's posing as someone with an occupation).
In one episode, the character of Col. Klink (as played by Werner Klemperer) 'crosses over' and there's reference made in the present-situation to his series, "Hogan's Heroes". This crossover doesn't make sense with the time, as it's made clear this series is set in the 1960s and 'Hogan's Heroes' was set in the 1940s.
Anytime Batman and/or Robin get into a situation, Batman almost always has the right gadget/medicine to get them through, or cure themselves. Even though they would never know the situation (or villain weapon) they would face. Also Batman always gets it out from one of his front pockets on his unity belt as if he correctly guessed it every time, even to put it in the front pocket for the mission.