When Zangrief is explaining to the official about the flash he sees when he moves a piece, his chess clock button is down -- later in the scene it is up (where it should have been because it was his move).
In Nate's match against Olivia Livingston, when he moves his Queen so it can be taken (hoping to confuse Olivia and make her waste time), he never hits the clock.
Hardison refers to the site of the chess tournament as the "second tallest building in the world". Establishing exterior and landscape shots, however, clearly show the Burj Khalifa in Dubai, UAE. As of its opening on 4 January 2010 and at least up to the original air date, this was the tallest building in the world, at a total height of 829.84 meters.
In Nate's next to last game, he moves the chess pieces with one hand, and pushes the button on the clock with the other. This is not allowed. You have to push the button with the same hand that moves the piece. In the real world Nate could have forfeited the game doing that.