Continuity: The Russian chess champion is shot and falls to the floor, taking the chess board with him. In the next shot, when Gibson Praise is looking up for the shooter, the chess board is on the table again and with some pieces neatly standing on it. Furthermore, when we see the shooting on video during the FBI briefing, the Russian merely nudges the board with his hand, which makes it spin a few times, dislodging some of the pieces.
Errors in geography: The Vancouver Science World (a dome like sphere) is seen in several scenes when the Smoking Man meets Krycek outside. In that particular scene, they were supposedly in the US
Continuity: At 00:35, you see the Russian grandmaster capture Kh8xRg8. In the next shot, Gibson plays Rg4-g8. These scenes are in the wrong order.
Continuity: When the sniper shoots the chess player, the player falls knocking the chessboard and pieces on to the floor. However, in the next shot the board is back on the table.