I'm using the same setup of Ubuntu 12.04 LTS with Evolution 3.2.3 running DavMail 4.3.2-2138 over IMAP, with OWA being 188.8.131.52
It seems to have trouble getting emails with attachments. There are some cases where once an email with an attachment comes in, Davmail will not get that email nor get any newer mail that comes in after the email with the attachment. I know this because once I've logged into the OWA web interface and moved the email with the file attachment to another folder instead of the inbox, Davmail works again and fetches other new mails fine.
Sometimes the evolution throws an error saying "cannot get information for message 123". However, most of the time it doesn't and is silent so it'll be a while before I realize that I'm missing incoming emails.
No logs yet. Please let me know if logs are needed or if the description is enough for reproduction and RCA.