Oil burning in cars is as old as the internal combustion engine itself. There are two major reasons for burning oil: mechanical problems with the valves or a seal issue with the rings. Valve problems generally happen at start-up with huge puffs of blue smoke, and can only be addressed by engine disassembly and repair or replacement. Ring seal problems will burn oil continuously as the engine runs, as oil literally blows by the rubber seal into the combustion chamber. This is the most common oil burning problem and can be addressed without any mechanical intervention.
Purchase an oil burning crankcase additive. Some popular brands are Rislone, Lucas, Wynns and Alemite. They are made to expand the rubber oil seals inside the engine and not allow any blowby of oil into the combustion chamber. These additives will not completely stop an oil burning engine, but they will slow it down.
Add one of these substances to your engine through the engine oil fill cap. Follow manufacturer's directions when doing so. One of the instructions will be to drive the car for a certain amount of time afterwards, and this direction must be followed specifically.
There will be a noticeable drop in oil consumption afterwards. However, this procedure will not completely fix the oil burning.
Take your vehicle into a certified mechanic and have the engine rebuilt. This will positively repair oil burning at the source, and is the only true way to fix this problem.
When using a crankcase additive, switch to a thicker grade of oil like a straight 40 weight. The thicker oil resists getting sucked past the rings of the car and into the engine, and the additive expands the rubber seals on the rings to stop burning even more.