Operation Not Permitted
imap(user): Error: chown(/home/user/mail/.imap/INBOX, group=12(mail)) failed: Operation not permitted (egid=1000(user), group based on /var/mail/user - see http://wiki2.dovecot.org/Errors/ChgrpNoPerm)
This means that Dovecot tried to copy /var/mail/user file's group (mail) to the index file directory it was creating (/home/user/mail/.imap/INBOX), but the process didn't belong to the mail group, so it failed. This is important for preserving access permissions with shared mailboxes. Group copying is done only when it actually changes the access permissions; for example with 0600 or 0666 mode the group doesn't matter at all, but with 0660 or 0640 it does.
To solve this problem you can do only one of two things:
If the group doesn't actually matter, change the permissions so that the group isn't copied (e.g. chmod 0600 /var/mail/*, see MailLocation/mbox)
Give the mail process access to the group (e.g. mail_access_groups=mail setting). However, this is dangerous. It allows users with shell access to read other users' INBOXes.