Under Lion my notes always synced using an IMAP account hosted at Media Temple, and it worked great. When I upgraded to Mountain Lion the notes were deleted from Mail, but they were NOT added to the new Notes app (and they were deleted from the server!).
Since the upgrade I've been unable to get any notes to sync with my Media Temple IMAP account, even though the settings are exactly the same as they were in Mail (in Lion). I've seen all the same error message in the console that others have (Sandboxd mdworker, deny file-write-owner, mdimporter, addNoteToRemote returned false, etc...). Any new note created is simply moved into a (local) Recovered Notes folder.
I switched my notes to my Gmail account and strangely it worked. Notes will sync via Gmail, but NOT Media Temple.
It should also be noted that notes created on my iPhone or iPad will continue to sync with Media Temple, so it seems as though it's a problem specifically with the Notes app and my host.
I encourage everyone to file a bug report with Aplle.