Yes, the tournament games count towards the game limit. But this first tournament is an exception because I need to test the tournament system as much as possible. Currently the only limitation is that a pawn can join only one tournament.
But the next regular tournaments will be defined with some restrictions. For example if a player sings up for a tournament but when the sign-up process is finished and he/she doesn't have enough empty slots for new games, the player will be removed from this tournament.