The problem is not physically hauling it away but rather disposal of the thing. Depending on where you live, old oil boiler/tank disposal can be a bit of a pain.

It looks like the boiler was disconnected and move into the corner so it would not be in the way. It looks like a gas fired boiler. If door ways are wide enough this can be hauled away.

Is the house currently heated by another boiler?

Yes - we have a heat pump up in the attic and I'm pretty sure you're right about it being gas. I can see where it probably used to hook in to the gas line. Someone said I should have a professional handle since there might be asbestos inside? But wouldn't it be fine if we don't open it or anything? Just haul it away?

I don't think these type of boilers had asbestos inside. But even if there is asbestos inside then you should be fine if you are hauling it out of the basement in one piece.