Hi Rebalancer, thanks for your feedback. Cards leave the table when they are played by humans during the second phase (the "chase"). Each round, when a human is attacked by the demon, they may counter it with an uncovered card from the center of the table instead of from their hand. If they do, regardless of if they were devoured, their card is discarded and the humans get a little closer to escaping the demon.
In practice, human players should be using and discarding the table cards most of the time, reserving the cards in their hand for when they're really in a bind.
I hope that helps!
Chris