Welcome to Fluther.
I’m assuming that electricity is included in the rental, and oil for the furnace is not. Therefore, it’s clear that you would be “expected” to use the oil heat, but it depends on the language of the lease whether that can be enforced / required or not. (If the place has hot water produced by the oil furnace, which is typical for an oil-heated house, then you’ll eventually want to use the furnace for that, anyway.)
Oil heat is definitely cheaper than electric, but perhaps not “to you”, based on your current agreement. Also, if the furnace is not used and you get very cold weather that freezes the pipes – including the heating water pipes (assuming a hot-water baseboard heating system) then you risk the entire structure in case the system freezes. (This could also happen if you go away for a weekend, for example, and simply turn off the space heaters since you won’t need them.)
Let’s say that you eventually use the oil heat.
In that case, insist that the landlord fill the tank initially, and you can be responsible for either leaving the house with a full tank again or paying him the estimated difference between a partially-full tank and what it will cost to fill. Otherwise, you’d be leaving him money in the tank that you probably won’t get back. (This is analogous to a rental car scenario. You rent the car with a full tank of gas and return it the same way, or pay the difference on the fuel gauge.)