The case sells darts for 0 gold to allow people to freely play darts with the nearby board. The darts still do full damage, however, and so from what I've seen there's nothing to stop you from loading up on as many as you'd like. Might be good to set them to 'No Damage' or some other such penalty.
I'm not sure, but isn't there a circumstance where the dart game specifically requires that you be able to do damage?
I would assume that the sort of player who makes a dart-focused PC is also the sort of player that we don't need to worry about exploiting incredibly trivial things. Perhaps time will prove me wrong.