The previous story, involving Mister Freeze, takes place in July, but the opening of this story states that it is April Fools' Day.
Bruce says that April Fools' Day is on a Wednesday. The series aired from 1966 to 1969, but April Fools' Day did not fall on a Wednesday during its run. The nearest one would be in 1964.
When Chief O'Hara goes to call Batman at the beginning, he picks up the receiver of the Bat Phone and the phone lights up red before the Chief presses the call button. Normally the Bat Phone will not light up unless the call button is pressed.
The newspaper articles Batman is reading don't match. If you look closer at the article next to "Governor Returns" it's about "Board Head to Speak"and the other is about a "Face lifting Job",
Eivol Ekdal tells Zelda to hide behind a trunk as he demonstrates that his chamber is bulletproof. However, he shoots at it unprotected.