2001 Honda Accord oil burning

An engine could burn oil for a number of reasons. First, note that the sign that an engine is burning oil is a light blue colored smoke coming from the exhaust (some people see it as gray). If your car is emitting this blue smoke then it is burning oil.
Now the causes. When oil enteres the cylinder, it burns, causing the blue smoke. It can enter the cylinder in a few ways. The most common are through the valve guides, in the case of worn/bad valve seals or guides, or as it slips past the piston rings, in the case of worn cylinders, rings or carbon build-up in the piston ring grooves (causing the ring to stick and not seal the cylinder correctly).
Valve seals are fairly inexpensive to replace. the technician would remove the old seal, check the valve guide for excessive wear (recommending corrective action if needed) and install the new seal.

Worn cylinders or piston ring problems constitute an engine overhaul, which can range very expensive to "yikes!" (depending on the amount of machine work needed). There are tests to determine piston ring/cylinder problems.

This might help: Generally, when the engine is started, if it emitts smoke for a few seconds and then stops smoking; the cause is most likely bad valve seals/worn valve guides.
If it smokes almost the entire time its running, it's most likely piston/cylinder problems.
There are some exceptions---though they are sort of rare.
Let me know what the technician finds out. Or ask any question that this info might bring to mind.