Then get some fungicide. Gosh. They do make a product similar to Stabil for diesel fuel. Several products, and some will kill the mold and other growths.
Personally I'd hit the start button if the oil were 5 or 10 years old. Unlike an engine, a furnace is a heck of a lot cheaper, and if it does get clogged, can be fixed relatively easily. And it probably won't.
The other posters have a legitimate concern : your old clunker could be leaking heat into the environment, increasing the thermal load on your heat system. If that were the case, I might think about removing it. Otherwise, I think you have the right idea : I suspect your furnace will last a long time if it is never turned on except maybe once a year.