If you have a written lease for the rental that you have you need to carefully read its terms and conditions in that this document controls the obligations owed to you by the landlord and vice versa in the absence of conflicting state law. If there is no mention as to whose responsibility in the written lease (assuming you have one) for the furnace's cleaning, it is the landlord's responsibility.If you have an oral agreement for the rental and nothing was ever said about the furnace's cleaning, then it is the landlord's responsibilty to clean it.If the written lease states that it is the tenant's responsibility to clean the furnace and you signed the written lease containing this provision, then it is your responsibility to clean it.