As Alfred enters the room to answer the Bat Phone, next to it on the desk is a black phone. When Alfred answers the Bat Phone the black phone is gone.
When Riddler asks his henchman to open the rare book safe, the safe bolts are already retracted, and the retracting wheel can only be rotated minimally.
When Alfred enters the study to answer the Batphone, it is facing toward the audience. In the next shot, when he picks up the receiver, it is facing the other way.
Early on in the candle factory & in the library, Matches' hat is orange, but it changes to blue to match his shirt when they return to the factory after capturing the Dynamic Duo.
When the Riddler cracks the rare book safe, the safe's bolts are on the outside of the door. Safe bolts are normally on the inside of the door.
When Batman and Robin are analyzing the chemical contents of the "universal solvent" wax in the Batcave, the spectral lines indicate nitrogen, uranium, and sodium. Batman and Robin then deduce that this indicates "NUS" (from the first letters) and then they turn "NUS" backwards to get "SUN" which gives them the clue they need. The problem is that the symbol for sodium is not "S" but "Na", so the conclusion is invalid.
When Batman is captured by Riddler and is in the van and later above the vat of wax, it's clear that the "compartments" on his utility belt are nothing more than ordinary yellow sponges.