Legal Question in Landlord & Tenant Law in Massachusetts

It depends on the terms of your lease. Read your lease very carefully to determine the answer.

If there is no applicable provision, then your landlord would more than likely be responsible for cleaning the furnace. Keep in mind that they may not want to clean the furnace and unless it is broken or unsafe, they don't have to.

If you are paying for oil, you might want to pay to clean it yourself in order to save by way of fuel efficiency.