I've actually wondered about this too as I was thinking of selling the site I originally signed up for Adsense for. I wonder if you might need to do a bit more than just remove the code (I assume they have some sort of file that links you with the domain name you signed up with) like write to let them know...
No, you should not. You need to apply if you are buying a site where adsense already implemented and you are not adsense publisher already.
photo200, that is incorrect.
You do have to re-apply if you were the original owner as well, and it was the site you used to sign up for. If it was simply a site you added after your original site, you just remove the AdSense script from the site.
Read seogrrl's experience here: [webmasterworld.com...] She had her account disabled when the new owners applied for AdSense and she had not reapplied for a new account.