When Batman finds Superman in the warehouse he picks up the piece of kryptonite, but it disappears in the next shot and reappears when he throws it at the TV.
When Superman crashes through the ceiling of the warehouse during his first confrontation with the Joker, the resulting hole in the ceiling is noticeable before he actually breaks through.
The U.S. president is driven away from the airport in a single black limo with no apparent police protection whatsoever. And this was after his plane was (briefly) taken over by armed hijackers! So much for security.
Lois Lane appears to be surprised that Bruce Wayne is handsome when he arrives on the plane. Since Batman's alter ego is a high profile playboy/bachelor who makes the gossip pages, how is it that the intrepid reporter has never seen a picture of him, or even seen him on television?
After Superman learns that Batman is really Bruce Wayne by using his X-Ray vision to see through the mask, Batman learns that Superman is really Clark Kent by planting an electronic bug on him and tracking him to his apartment. Since Superman has heightened senses beyond those of anyone on Earth, he should have been able to hear the tracking frequency emitted from the bug.