The Joker throws a gun through the windshield of his car, but in the next shot, the windshield is intact.
When Harley and Ivy get into Ivy's pink car to escape, after they pull out of the driveway, ramming the Joker's car out of the way, the scene changes to Batman and the Joker fighting. Even though Harley and Ivy just took off in Ivy's car, it can be seen in the background of Batman fighting the Joker in the very next scene,
After blowing up the car of the guys who harass them, Harley and Ivy drive away, but their headlights aren't on in this shot.
The right shoe of the henchman lying unconscious in the background, while Batman struggles with his partner, is colored brown like his pants. The other one is colored (correctly) black.
In the first scene, when Batman chases Harley and the Joker, they run through a stoplight that has the red light where the green belongs and the green light where the red belongs.
When the girls throw Batman into the toxic pool, he is shown to open his mouth underwater. Considering the circumstances, this is an extremely unwise thing to do, and would mean panic and a loss of control quite uncharacteristic of Batman.