Currently, the rental unit is empty so I just shut off the furnace and the water heater to conserve oil, and frankly because there's no need for hot water and heat (currently not freezing).  

You are also obliged to have heating in the unit, but that doesn't mean you need to have the gas bill in your name...?

That seems to be the most fair for the water, but since I think the gas is used not only for heat but also for stoves/ovens, I'm curious to see if anyone has another way of splitting this out.Any thoughts?

Here's the situation:  I've got a single story SFH in southern AZ.  The house has natural gas lines running to the gas furnace, the kitchen and the laundry room.  The only thing hooked up to the gas line is the furnac...

Trulia has a heat map, neighborhood Scout has some cool tools.

Once there, they turn up the heat with very polished and professional "motivational speakers" who know how to work a crowd. 

It currently has heat only( original 50s gas furnace).

Good point about the heat.  

According to numbers I've received from the listing agent, the annual heating/oil expense is just shy of $11,000 a year.

One heating system (owner pays).