Here are some opinions and ideas for you to think about... Just so you know that I'm not just shooting from the hip.. I worked in the heating and air conditioning field as well as attended school.I need to get my heat exchanger replaced as the old one is failing. It has a five minute burn off of the oil applied to a new one for installation.
Here is a diagram of a typical forced air furnace. It is gas fired and I assumed yours is also. There are as many different types of furnaces as there are cars.

1. If your heat exchanger is failing,, (the silver colored part in above picture.. top,,,,, back) maybe it is time to replace complete furnace. Heat exchangers usually last a long time. When they go, then all your other components are close behind one at a time.
2. Removing one is not an easy task. (varies by furnace) so often times it is easier to remove furnace from installation and disassemble , and rebuild in a stand alone position. I don't know your particular furnace or circumstances. Consider the cost of a labor intensive rebuilding compared to a new furnace replacement.

3. If done in that method, then have your contractor fire test your rebuilt furnace outdoors and burn off that oil film outside. He/she may tell you some BS that it is not necessary. To you it is a legitimate concern. Others with chemical sensitivity, @lazy gardener and @CarolinaSunshineFlock IT IS IMPORTANT.
4. The fire testing will burn off the oil film that is from stamping/manufacturing process. It is on both sides of the heat exchanger metal. @aart . The interior of heat exchanger will vent up the chimney. The exterior portion when heated and burned off will enter the living area, if done inside and already installed. Think of the surface of the metal under a microscope as sandpaper. A very small amount of oil may still be present on the surface until burned off.
5. A new furnace should be fire tested at the place of manufacturing to make sure ALL safety devices and controls are working properly. That test would burn off any oil residue.
6. Your service personnel may tell you that they have no way of firing your rebuilt furnace outside. If they are half as good as I, they should be able to figure out how. Sometimes done to make sure a unit is operational, especially when purchasing a used model before handing over cash.
Here is the danger of a failed heat exchanger. Pix is from online search.

A Carbon Monoxide detector should be in every home. Here is a pix of mine.
As well a smoke detectors properly placed.
IT IS THE SMARTEST LIFE ASSURANCE AND SO REASONABLE.