The Bat-Signal shown in the sky in the series 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 is hardly ever seen in the next two seasons; in fact, it is barely used. It nearly disappears and is barely mentioned throughout the rest of the series' run; this is never explained and it is left oblivious.
In episode 7, Alfred refers to Robin as Mr. Ward, and not Mr. Grayson. But the series often refers to Dick Grayson / Robin as "Bruce Wayne's young ward." It
is not a reference to the name Burt Ward.
Several times over the course of the series, Bruce refers to or addresses Mrs. Harriet Cooper as "Aunt Harriet." Strictly speaking, she is no relation to him. Obviously not, but she is 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 that Bruce would use it.
At times during episodes in the series, Bruce Wayne and / or Dick Grayson (and later sometimes Barbara Gordon) are able to change into (or out of) their superhero costumes very quickly (or would not have enough of an area to do it). This is a common cliche in crime-fighting superhero incarnations by their artistic license.
Any time when Batman and Robin are climbing up a wall in the series, you can sometimes see the strings (at the bottom of the screen) 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 has been explained (even by Adam West and Burt Ward themselves) that it was done purposely to have some fun for the series' adult audience.
Quite often in the series 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 of the series, during fighting scenes, Robin's (Burt Ward) stunt double can be spotted.
The road sign just outside the Batcave in the series says "14 miles to Gotham City". Wayne Manor, which
of course sits above the Batcave, 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 the series.
In some episodes of the series, Alfred is correctly recognized by villains (or allies) as Bruce Wayne's butler. In other episodes, the same villains recognize him as a different person (always when he is posing as someone with an occupation).
In one episode, the character of Colonel Klink (played by Werner Klemperer) "crosses over" in a window cameo and there is a reference made in the present situation to his series, "Hogan's Heroes". This crossover does not make sense with the time, as it is made clear that this series is set in the 1960s and "Hogan's Heroes" was set in the 1940s.
Any time that Batman and / or Robin get into a situation in the series, 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's weapon) that 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.