I just fixed the Notes vs IMAP-problem at my system. Lucky shot, but I can sync Notes from iPhone to Macbook Pro running Mountain Lion back-and-forth via my IMAP mail server (which is a Mac Mini running Snow Leopard Server). (I don't use iCloud)
I shall describe what I 'switched on' at my server. Maybe it's of some help for some of you.
My problem was:
Syncing notes from/to my iPhone and iPad worked 100% over my IMAP mail account with my MacMini SL Server. Fresh Notes always did show up in mail at my computers, until I upgraded my MBP to Mountain Lion. After this Notes came up empty and Notes kept asking for a password.
What I did:
In Server Admin at my Mac Mini server, in the Mail settings, under Advanced there were two checkmarks at the IMAP/POP Authentication setting. Checked were: CRAM-MD5 and APOP (Pop only)
UNchecked were three other settings: Kerberos, Login and PLAIN...
I checked ALL three settings; tried Notes and it worked. Immediately showed all my notes from my iPhone and iPad; setting new notes and deleting notes back-and-forth worked too!
After this, I unchecked the settings one-by-one and the setting 'Login' was the one I needed to have checked to have notes working via IMAP.
So now Notes is running under Mountain Lion via IMAP (as I said I have iCloud switched off) with my Macbook Pro. My older Mac Pro is still at Snow Leopard (with the notes being part of Mail) and this is updating fine too.
Hope this helps for some of you. At least it's not a Mountain Lion bug, I guess...