I would think if it's something you can buy at the super market, the tenant; if it's something specialist, then the landlord. But that's just my commonsense take...
Yep Bunnings supply them just need to unscrew the top and pop in a new filter $8 each. Its not the price of the filter is the price of the call out for a handyman to install. Wished i ripped the thing out now. Its a bit like the dripping tap. $2 for the washer $120 for the plumber.
Yep take it out when changing tenants, for now its not worth the puss & pain over such a low deductible cost. If you have to call someone out to do it get them to also do any other minor repairs at the same time (leaking washers etc) to make use of possible call out fee rather than having them back in a few weeks.
It is a consumable so it should be the tenants but agree with the others to have it removed as soon as you can. Too much of a bother.