Whether or not they burn oil with age depends some on the model (especially if you drive them hard or at high revs) but how they're taken care of is the main factor.

Assuming you're running the correct oil, and that the engine isn't running hot, it's mostly likely worn rings or the old, hardened seals that's causing the oil use. Normally, unless the car is abused or run on very low oil, a Honda's oil use (if it's substantial) will be due to seals, not worn piston rings.
If it was the rings, nothing you can do w/out replacing them (not cheap). If it's the valve seals, you've got options - replacing them isn't that expensive, especially if the shop knows what they're doing (meaning they don't remove the head), or you could try a product meant to clean out sludge and/or condition seals. There are high-mileage oils you can buy that are designed to do both, or you could try two separate products.

I'd also take a look at the PCV valve to make sure it isn't broken or clogged.