I am stunned by the number of people who play right into lazy/corrupt ex-tenants' hands by enabling them. Don't send on, send back.
Don't ge tme wrong - if it's just a few instances, and it very quickly drops off, that's different.
We made up labels, saying "Return to sender - occupant has moved away. Remove from database" and most of the post trickled off pretty well after that.
Some persistent posters didn't get the hint for a while, but we'd also kept a really low-tech database (A4 lined pad with date and sender) of any sender information we could find without opening the envelope (company logo or postcode, for instance). After a while, if we wanted to, we would add to the labels "you're been told xx times".
A previous landlord did the "forward my post" thing. We refused. The agency tried to give us grief, but we pointed out that part of our lease that said we weren't to engage in any business, and that providing secretarial service, even if unpaid, is exactly that. Turns out most of the post was from the bank, whose mortgage wasn't buy-to-let.