What could cause a bad oil-like smell every time the heat is on?
Your question concerns me. I'm a general contractor and have had many new heating systems installed for customers and have NEVER had this complaint from anyone. As a matter of fact, I just had a complete change over from oil to natural gas done at my private home two weeks ago. I have never smelled any residue from cutting oil used to thread pipe. My sub contractor is very conscientious about cleaning all excess oils and pipe joint compounds. Also, before any gas can be turned on, an extensive test of pressure and inspection of all black pipe and outlet fittings is done. There is absolutely no reason to have any fumes.

The other reason for my concern is that you smell this when you call for heat. My first concern is that you may be getting some exhaust or carbon monoxide leakage. Do you have CO and explosive gas detectors installed? If not, get them installed immediately! If your contractor won't take your complaints seriously, then call the gas utility and ask them to check it out. Tell them you suspect a leak, and they will be there in a heart beat!

I should ask, what kind of furnace unit/heater are we talking about? Vented, unvented? Give us a bit more info and maybe we can narrow down the possibilities.