Ideally you should speak to your lodger about this before you leave. For example
- If they get a lot of mail, tell them that you cannot be responsible for forwarding this and that they must arrange for a postal re-direct
- If they only get a modest amount of mail, ask them to leave you some address labels with their new address on and say that once these are used up you will return post to sender (an idea suggested by Lynda who commented here)
- Say that you will forward post if they leave a forwarding address for a limited period (say a month) and that after this post will be returned to sender.
What you do not want to do is get in a situation where you are having to constantly forward post on a long time after the lodger has left. That is not fair on you.
If you decided not to forward mail any more, or if you have not been given a forwarding address you should NOT open or destroy the mail. It is not your property and you are not entitled to deal with it. If you cannot give it to its owner it should be re-posted marked ‘return to sender’. Or ‘not at this address’.
The advantage of returning it to the sender (if the address is not on the envelope the mail staff will open it to find out their address) is that this will, hopefully, inform them that your lodger is no longer there and that they should update their records.
If you have house rules for your lodgers, you could usefully put something there about you procedure for dealing with this.