if the oil tank is empty, will a furnace continue to draw electricity

The furnace should have a flame failure sensing system on it that shuts the burner and ignition down after 2 or 3 failures to see a flame in the combustion chamber which would result from no oil. This should be a manual reset and would not run again until manually reset. If it was actually trying to refire, turn off the power and have it checked by a qualified HVAC technican. While that could cause a slight rise in your power bill, I doubt it would be very high. I would suggest you look at the bill and a pervious bill and see the difference in actual KW usuage. It could be a increase in $ cost of usuage rather than actual KW usuage.