One thing to try is to replace the PCV valve. If they stick open, they'll
suck oil in a way that's indistinguishable from rings or valve seals.

Essentially, you bought a worn out car. Every mile you put on it is a
miracle and I'd just be happy it still goes. I'd probably not spend
anything more then normal maintainance on it and when it dies, have someone
haul it off to the junk yard.

Mechanic probably told you you needed "valve seals" and not a valve though
he might be recommending you get the head done which reconditions the
valves and replaces the seals. I'd just keep a case of oil in the trunk and
add a quart every fill up. If it's not leaking, it's either valve seals or
piston rings. If it's the rings, then the engine's really worn out. While
Honda engines can last forever (almost), if some previous owner wasn't good
about changing the oil, your engine might have excessive wear.

I am a new to the "honda owner family."  I bought a used Honda Accord
LX with 165,000 miles on it two months ago.  I have since noticed,
unfortunately, that it is burning about one quart of oil between every
time I fill up with gas.  A mechanic told me it was probably an old
valve that had gone bad.  He did not quote me a price for repair, but I
can imagine it will be expensive.  What do you all think?  With my
Accord being 12 years old and having 165,000 miles, would it be worth
it to get the valve replaced just to avoid the nuisance and potential
hazard?