This sounds horribly unethical. The coins don't belong to the OP. The original owner might have a backup and might try to move the funds once they notice that their paper wallet is missing.
Let's say few weeks and months pass and nothing happens.
Would you simply burn the words, give it to police, or maybe give it to some poor people?
There is always possible scenario of someone finding private keys of satoshi coins, any many other keys for inactive coins could be find by other people, so I don't see any difference with lost coins.
Lost keys = Lost keys.
The point is that we don't know for sure if the original owner lost the keys completely or just one copy thereof. It's not like physical cash, which if unclaimed can be reasonably assumed to be lost after a certain period of time.
For example the original owner may not be aware that one copy of their key is missing, or may think that it was destroyed. Sloppy, yes, but they might still be hodling another copy of the key thinking to cash out when Bitcoin hits $1m. So I don't think there is any time limit as to when this wallet can be assumed to be lost.