Hey I thought that I was the only one with this problem!
I have a '99 LX (AT) with 132k on the clock. I have been putting in a quart almost every fillup. I had the timing belt done on time and changed the gasget. There are no oilspots in the driveway and no obvious leak coming from the engine. The tailpipe is clean. I'm told by everyone that Hondas have no problem at all until 1,000,000,000 miles. HA! My mech told me that old cars do this and that I should just be sure to put in oil when I need it.
I started using high mileage oil, whatever that is, and it has helped somewhat. Now I only need to put a quart in every other fillup, but only if I put high mileage oil, whatever that is, in then too.
We decided to take a trip to Spring Training in FLA last March from NY and I was worried that the old silver shost would be toast somewhere along the line, but she served us well for the 3,000+ mile trip. Got to see Santana pitch too. One night, I had to get on the interstate and had to floor it and, as I did, I noticed that smoke had come out the back (pretty bad too). Since then, I've taken it a bit easier and seem to add oil less.
Mrs. Fudbutter's solution is simply to get a new car so I took her to the car show and told her to pick. She liked three cars, two had a BMW emblem on them, one had a Mercedes. My solution is to see if I can run a '99 Accord for 1,000,000,000 while adding oil every other fillup.
I will follow this thread with interest.