Here’s one:

The Joker crashes a party and clearly is on a homicidal track, bringing along a number of henchmen to help him out.  He threatens Rachel Dawes and then Batman appears and everyone was kung-fu fightin’… So The Joker tosses Rachel out a window and Batman dives for her and, I suppose by the laws of Superman physics, manages to catch her and slow down their fall enough to keep them both from having their bones crushed when hitting a car.  Never mind this physical magic (so magical that even Rachel wasn’t out of breath enough for her to make her singular semi-comical quip in the entire movie), but the next scene we see… Batman standing atop some building looking all… Batman-ish.  Nice shot but, uh, what about the party?  The one with The Joker, some goons, a bunch of guns, and a lot of innocent people whose lives were being threatened.  I supposed Batman just figured The Joker would get bored and leave because we all know the party’s over once Batman leaves.

Here’s another:

The Joker is in the interrogation room, and someone who shall remain unnamed un-cuffed him, and he’s left with a cop standing guard on him.  He’s taunting the guard, obviously to rile him up so he’ll start a fight with him and presumably overpower him to expedite his escape.  The next thing we are shown in that police precinct is The Joker now has that cop overpowered with a sharp object (probably a piece of glass).  So when can the doors in interrogation rooms be opened from the inside?  Who’s genius idea was it to leave a homicidal maniac un-cuffed in a room with a single guard?  Something was missing here and it was a gaping hole which felt that way.

Yet another:

Just before The Joker and some of his goons pour gasoline onto the huge pile of cash, we are shown the Chinese banker tied to a chair at the top of the pile.  The camera pans a bit and we never see him again.  It’s obvious he was set on fire atop the blaze, but it’s never shown or even visually asserted, other than the quick shot of him that is very easy to miss.  Probably there were scenes of him on fire that were too intense for a PG-13 rating, but then why even have him visible in a glimpse in a shot like that?  And why not have the Joker comment later about setting him on fire?  It’s like the editing wasn’t fully thought through for that scene and then it’s quickly forgotten.