Landlord-tenant laws are state-specific, and given the number of states it's impractical to scan all of the laws, but based on a reading of a handful of such laws I doubt that there is any law requiring landlords to pay the oil for a rented house. (The matter would be different if there was a multi-unit building with no individual control over temperature, thus pooled fuel usage).
It's not clear to me what you assumed the agreement means, where it says "N/A". Perhaps you believed at the time that the place had a different heating system, and you relied on that assumption. In that case, you might be able to go to court and have the contract voided, and you could pick another place to live. If the "options" are specified so that some things are assigned to tenant, some to landlord, and some are N/A, that would especially lead to the reasonable belief that there was no oil heat in the house. But if the only indications were "landlord" versus "n/a", then you could interpret "n/a" as meaning "not the responsibility of the landlord". Analogously, if the agreement only lists "tenant" and "n/a" then a reasonable interpretation would be that this means "the tenant pays" versus "the tenant does not pay". This reasoning would also have to survive the alternative interpretation that the tenant pays for everything, except that n/a means "there isn't one of these". In other words, the meaning of the term might be determinable from the overall context of what's in the agreement.
Since the house does not come with a full tank (as with car rentals), the question of what to do with the residual oil at the end of the lease should also be specified. Unlike gas or electric, you're not just paying for actual consumption, you're paying for potential consumption, and you would have an interest in the remaining half-tank at the end of the lease. You could just walk away from that investment (pumping it out and taking it with you could be illegal, since the stuff is kind of a contaminant), or you could have an agreement where the landlord buys the oil back from you, but that should be specified in the agreement (and I assume it isn't). This kind of consideration could support a claim that you reasonably believed that there was no oil system (if there were, there would be some term relating to your interest in the residual oil), or even a belief that the landlord would pay the cost of the oil (since he ultimately gets the remaining oil at the end of the lease). You attorney (hint) should advise you how to approach this.