Continuity mistake: After the prologue, you see the Montague boys in the yellow car. If you look at the Montague sitting in the back of the car, you can see that his sunglasses are on his forehead; a few seconds later he puts the glasses down over his eyes. In the next shot, his glasses are back on his forehead again.
Continuity mistake: Tybalt's tattoo changes constantly. When he's watching Mercutio dying on the beach, it's a plain simple heart on his chest. When Abra is forcing him to run away to his car (in the same scene) it is not there at all. Then when Romeo is chasing him in his car and Tybalt's car rolls over, as he crawls out, it is much more intricate, with black lines and fire. It also changes when he gets shot over the fountain.
Visible crew/equipment: Near the end when Romeo is having a shootout with the police at the church, look closely at the door to his right. There's a bit of wiring or something there. An officer fires a shot toward Romeo and you can then see that the wire is actually the flare/effect for a bullet hitting the door. (01:40:00)