A landlord should go inside the rental every month with a filter and at least supervise its installation. This gives the landlord a chance to look around and take inventory of any pets, junk cars and squatters. Filter changes should never be the responsibility of the tenant.
Jake, is this in your lease you're changing the furnace filter every month? Personally I do a (at least I do now) inspection quarterly of the property. I really don't think I should have to montly look into the house (unless I suspect somethings not right - too many people, etc. - I live nearby so drive by the house daily).