While not good to run your vehicle while smelling a burning oil smell, oil on hot parts will usually not catch fire and cause your car to go up in flames. You'd need a very hot manifold or exhaust to cause it to do this, plus you need a lot of oil. Almost always, oil will just cake on a manifold. You'll get some smoke and smell, but nothing worse. It is annoying, that's for sure.
Most vehicle fires are either caused from gas leaks or electrical issues causing insulation to burn. Once a "flame on" situation occurs, all sorts of ugliness ensues.
While almost always oil leaks are not dangerous, it's not a good idea to run a vehicle around with a severe oil leak. You're just wasting oil, for one thing. Plus the environmental aspects of it. As a swag, I'd bet over 1/2 of the vehicles on the road today have some sort of oil leak. Most of them are so minor, you'd never notice it. You ever look at an older car and see gunk buildup on the lower part of the engine? This is most likely caused from oil seepage, which then collects dirt and grime.
Your best bet is to determine the amount of oil which coming out of the engine. When I was in the Army, we had a system for telling when an oil leak needed to be fixed. There are considered three classes of leaks:
- Class 1: Film of oil (or whatever liquid) is present - Continue to observe the leak; no action needed (Note: Brake and fuel Class 1 leaks are automatic deadline)
- Class 2: Drop forms, but you do not witness it drip; continue to monitor; prepare the vehicle to go to the 3rd shop for repair; vehicle is still usable
- Class 3: You see it actually drip to the ground - Vehicle is deadlined and can no longer be used until the problem is fixed; get the vehicle to the 3rd shop for repair
While these don't exactly apply to life in the real world (meaning, unless you've been a mechanic in the US Army, you wouldn't classify them as such), they do have merit and you can apply them to how you treat your vehicle. Leaks are not good and bad leaks are even worse. You should plan to get your vehicle fixed at your earliest convenience. Just ensure liquids are topped off in the meantime.