As mcasser posted Honda considers 1 quart per 1000 miles normal for the K24 engine dating back to 2003. I never had an engine that consumed oil at that rate and I don't consider it normal. Honda can claim what they want.
Hopefully components will seat and consumption lessen, but it should be noted that Honda has had some issues with these engines related to faulty oil control rings dating back to 2003. Apparently this isn't a pervasive problem, but if its your vehicle... You might want to document your current consumption with Honda now and again if it increases. It probably would be wise to keep your oil change documentation.
It probably would be a good idea to do some research here and on some of the Accord and CRV forums related to oil consumption and faulty oil control rings in this engine. The NTSA complaint database is another good source. I think a class action suit was brought against Honda for this problem for model year 2008(?).
I have a 2003 K24 engine with only 55,000 miles. I have never consumed any oil,knock on wood, but check it weekly. Some unfortunate owners experienced consumption first at 60,000 miles which worsened over years to the point they were burning a quart in 650 miles. Can't imagine a catalytic converter could function in that environment for long.
