DJMala

I'm no expert, but if you're actually burning oil (i.e. blueish smoke coming from the tailpipe) the most common cause is simply old age. How many miles do you have? Over time parts wear, gaps get wider, and eventually oil starts to slip past the piston rings into the combustion chamber. There are some stopgap measures, like using thicker oil, but the only real fix is to rebuild or replace the engine.

If you're just smelling it and the oil level keeps dropping, it's possible you could just have a leak that's dripping onto a hot engine part. The rear main seal can drip on a hot transmission housing, or valve covers can leak onto the exhaust manifold. The oil cooks off, giving you that burning smell. You may even see a little smoke, which will seem to come from no place in particular in the back of the car as you drive. Watch for spots on the pavement after the car has been parked in one place for a while. This will generally be cheaper and easier to fix, although something like the main seal can get a little expensive because just getting to the part can be labor intensive.