Neth's dropped the lowdown on rested XP in the expansion. In short, for level 60 characters, it's going to reset, and then restart.
In a little longer, she says that you can't earn rested XP at 60 right now in preparation for the expansion. After you install Burning Crusade, your level 60 characters will not have any rested XP at all. But they will start earning rested XP as normal as you go on your way to 70. So if you want to be really crafty, even if you're planning to sit out the expansion insanity, you might drop the expansion on your computer, and then make sure your level 60's are in an inn while you stay logged out for a few days. That way, not only will the servers be back to normal when you get back, but your toons will be ready to go, too.
As for lower level characters, I'd imagine that rested XP will work the same whether you have the expansion installed or not-- if they have rested XP saved up before you put the discs in, they'll have it afterwards, too.