What to do if you run out of Fuel Oil - Beckett Oil Burner Problems

Hate to say it but you need a service tech. Oil furnaces are very fickle things especially if it has not been serviced yearly. It could be no oil or if tanks above ground the oil could be jelling in the lines especially if there exposed outside. Could be clogged oil filter in filter housing or on pump or nozzle. Could be bad cad cell dirty heat exchange. Could have a cracked heat exchanger. Its gonna take a tech with the proper tools to figure it out.

For clarity, a Furnace produces HOT AIR for heat while a boiler produces HOT WATER for heat. Just remember...boil = water. You said your "furnace" does not produce hot water for a shower and because furnaces do not product hot water unless they have some model that do???? Usually people with a furnace have a seperate hot water heater. People with boilers though can produce hot water for use as both their sinks/showers hot water and their heat (by heating water that goes through baseboards) in their house. I don't believe any furnaces produce hot water most of the time a furnace just produces hot air for heating. So figure out what you first have.
Having said that, I'll assume you have a boiler like I do and my Weil McClean(sp?) stopped working a while back and wasn't turning on to "boil" any water. I took apart the burner and there is a light sensing photo resistor as part of the controls. If this phto resistor goes bad, the boiler will not start. The resister is cheap, I think it was $8 but you have to know what you're doing to change it. So you may be better to call someone who can work on such equipment.
Boiler's themselves are not overly complex. Home heating oil is the same diesel fuel that you can buy in a gas station only the government has "oil" companies put a RED die in home heating fuel which is usually cheaper then Diesel fuel because diesel fuel is taxed to death. The reason they do that is so that if you try putting RED tinted home heating fuel in your truck and you get pulled over (because you're a trucker with and 18 wheeler and they typically do inspections of these trucks) you will be a huge fine if they see you're running home heating fuel and NOT paying your taxed by purchasing Diesel fuel. A little bit of background so you know the fuel you are dealing with here. So it's dieslel fuel without the tax you run in your boiler. Gasoline on the other hand is VERY explosive as you know, but diesel fuel (if you're ever tried to light it) takes some coaxing to get lit. When it's cold out, diesel fuel is very hard to light and that's why trucks use glow plugs. You don't need those in your home though.
But because diesel fuel/home heating oil is hard to light, it's sprayed as a msit into your boiler, so that it can light more easily.
But because it is a fuel, you should know what you're doing when messing with it. FInd out what you have, and then have someone work on the issue if you haven't already. I'm guessing you have had it fixed by now?

Furnace needs vaccuming.

if i understand your problem description correctly when power is on but the furnace is not running you can smell burning dust and feel heat? first thing i can think of is possible a heater relay has gone bad keeping the heater on all the time but no fan...