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Exactly how far you need to go depends on whether you have steam or hot water heat. I'll get to that.

This will assume that you do not have a generator and that you are, in fact, out of oil.

Domestic plumbing. If you are on city water, turn off the water at the house entrance. Open all the faucets all the way, and flush all the toilets. Open any drains or hose bibs on the domestic water. Turn off the power to the water heater and drain that. Get some RV antifreeze. I say again RV antifreeze. Do NOT use automotive antifreeze. Plunger the toilet bowls to get as much water out as you can, then add enough RV antifreeze to each one and to the tank for the toilet to make about a 50/50 mixture. Then add RV antifreeze to all sink and shower and what have you traps.

This will give you a good chance that you won't have frozen pipes. It is NOT a guarantee. There may well be low spots in the plumbing which will hold water. They may freeze despite your best efforts.

So... to turn the water back on, close all the faucets and drains and what have you. Turn the main valve on, but listen for any evidence of water leaking and be prepared to turn it back off again. Go around and open all the faucets and let any air out. Particularly the hot water. When you get no more air, turn the power for the water heater back on.

Now the heat. For either type of heat, turn off the power to the boiler.

If you have steam heat, unless it is an extended emergency, the basement where the boiler is is unlikely to freeze, so you don't need to do anything. If it is an extended emergency, open the boiler drains and any drains on the wet returns. There's no need to do anything else.

If you have hot water heat, this assumes you do not have glycol in your system. You will need to drain the system completely. Open any drains you can find in the basement. Go around and open the air release valves on all the radiators. If there were any radiators which were turned off or turned down, open the valves fully.. If the expansion tank is mounted with the water connection up (air on the bottom), disconnect it and empty it. Close the connecting valve to the domestic water.

To restart the system, you will need to refill it and purge all the air. Take your time. Like the domestic water, there is a chance that not everything will have drained, and you may have some frozen pipes despite your best efforts.

Or make sure you have a properly connected whole house generator, enough fuel on hand to run it for a couple of hours a day for a week or so, and enough fuel oil in the tank...