Can a US resident cross the border with old furniture from his/her Canadian household?

If the items are you personal property, for your personal use, then there is no problem. Things to avoid? Sure, don't bring 15 laptops, 7 sets of golf clubs and 25 DVD players with you claiming that they are for your personal use. Things like that are obviously for import and resale only so avoid that as well as the standard limits for cigarettes and alcohol. Other then that, as long as the stuff is yours for your personal use then you will be fine.