It would depend on who put the refrigerator in the apartment. If it is yours and you bought it with a water dispenser, then you have the responsibility for it. Of course, if the tenant purchased it, the responsibility is theirs. However, if the refrigerator came with the apartment when you took ownership, this would be a disussion that you would have to have with the tenant. If they do not want to have the responsibility for changing the water filter then you should disconnect it and they should buy a Brita.
Even if it is the tenant's responsibility and outlined in the lease, if you are concerned about it you should check it and change it. I would do it with other filters and lubing the garage door or other routine items in one visit.
If you want the tenant to do it then put a dollar amount on it and have this is writing similar to this: The tenant can choose to change the filter or pay $65 every 6 months for parts and labor to have this done. If the tenant chooses to change this filter they need to provide the landlord with a receipt and verification of the change.
It is more likely to be done if there is a charge attached to it. Then if they don't want to do it they know they have the option to just pay for this. No need to fight and argue about these things later. Keep it clear and in writing.