I have nearly the same problem with IMAP on iOS 7. I have my own rootserver hosting my emails and domains. On the different domains I have some email-accounts. Hosting is configured with Plesk, normal CentOS based.
I configure these email accounts as "normal"/default IMAP accounts in iOS 7 on my iPhone 5.
So far so good. One of these accounts - exactly this and only this one - cannot connect or download new mails after a time between 16 to 26 hours. This means the first 16 to 26 hours I can connect to the server, can read and download mails. Then suddenly iOS 7 cannot connect to the server anymore (without changing anything!) or cannot download new - but existing - emails from this single mail account. The other accounts work normal!
I can remove the error by deleting the one email account from iOS 7 and re-installing it. Then it works for another 16-26 hours...
The only difference between the mail accounts is the username@domain.de. No other difference. It just worked all fine with iOS 6.3 so it IS definitely a problem with iOS 7! No difference with 7.0.2.
I hope Apple will fix the IMAP problem soon. Didn't they - or the thousands of betas - test IMAP with multiple accounts? They act like beginners not like pros...