What happens to all the coins in lost wallets in the Bitcoin system?
Those coins remain usable to anyone who is able to obtain the private keys to that wallet. If they are not found ever, then the coins are considered as good as lost and nothing happens to them.
If there are only a finite number of coins, won't they become increasingly scarce as people lose them over time?
Yes. This in turn makes every bitcoin owner more and more cautious about the number of coins that they own and secure their own assets. Lost coins end up raising the value of the coins that other people own and that is why one must guard their coins with proper measures or end up like the people who talk in chatboxes about their old hard disks with password of lost wallets.