There are some very useful Reddit posts on this Reddit comments on this topic, and one AeroPress user states:
“After I press, I let the coffee cool a bit, I peel off the filter, give it a rinse and let it air dry on the filter cap. I usually reuse it 10-20 times before I throw it away. I don’t have a set number of reuses, I just keep going until it looks a little worn…”