After Neal passes out in the vault, he is laying on his side against the wall. A few minutes later, when Peter is trying to get him to breath, he is flat on his back, and about a foot away from the wall.
After Neal excuses himself to use the bathroom and is on the stairs looking at the photographs of Peter & Elizabeth, he's standing directly in front of the photographs when Elizabeth asks if everything is all right. When they cut to Elizabeth's POV looking up at Neal, the photographs are gone. They reappear on the next shot over Elizabeth's shoulder looking up at Neal.
When Neal and Peter are locked in the vault and searching for the kill switch, Neal moves the boxes on the shelves. However when Peter is reviving Neal the boxes are neatly organized again.
The email Burke brings up on his computer is not information on Avery, its an email about Student loans.
The sound of a pump action shotgun being racked is heard - but the shotgun being used is an over/under.
The sound of a plate shattering is heard when the clay pigeon is broken - the sound of the pigeon breaking would not be heard over the report of the shotgun, nor would it sound like a plate shattering.
When Neal sweeps the pieces off the chess board one ricochets forward, revealing the screen protecting the camera.
In the chess game shown at the very beginning, the black Bishop is on the same diagonal as the white King, which is covered by the white Knight. Caffrey moves the Knight opening his King to a check - an illegal move. In the next bizarre move Mozzie takes the white King. Even as an allegory it doesn't make sense.