Why would a car burn oil when it does not leak or blow blue smoke?

More details would be helpful. However if your car is using too much oil it has to be either going on the ground or thru the engine. Please try to clarify and give more details for instance how much oil is it using? Has the car sit for a long time? Is there any sign of oil on the engine itself? Is there any signs of oil underneath the car look well at the underside of the floor board? , EzForJesus

It could be that the piston rings have rubbed the side of the cylinder and caused wear ,you may have to have the engine rebuilt. You can't always see the smoke. It may only smoke when you are traveling at high speeds and disperses in the wind.