Deliberate mistake: When Constantine comes out of the library after speaking with Gabriel, he stands beside Angela and none of them cast any noticeable shadows on the door and walls behind them. When it cuts to a shot above them, shadows are more being cast behind them. This is probably because of different light settings on the set.
Deliberate mistake: When Constantine and Angela are driving while talking to Beeman, as Beeman says "But that doesn't mean that we don't have faith in you", the lights in the car suddenly go off. Revealed by the DVD commentary, it is the cinematographer's lights going off by mistake.