When Finch and Root's elevator finally starts moving, a camera shot through the shotgun blast hole shows the arms dealer and the room moving up (ie the elevator is going down). But this would require the outer door to be traveling with the elevator, instead of staying on its floor. We should have instead seen the back of the outer door through the hole.
Root escapes a gun fight in a standard NYPD car and while the car takes multiple hits there are no bullet holes or other damage. Even later when they are seen getting out of the car and walking around behind it there are no bullet holes or damage.
The wall Root breaks through to get to the pay phone, is part of an old subway station. And yet the wall is made up of unsupported tiles that are less than an inch thick.
A large number of heavily armed men have a street in NYC closed off and are making loud speaker announcements. And no one including the NYPD seems to notice. Even when a major gun fight starts. At the same time a dump truck T-Bones a car into a building and there is no interest by the public or the NYPD.