Could it be some (older?) webmail clients that use pop3 instead of imap?
I wouldn't expect a webmail client to hold a pop3 connection open across multiple web requests. We have standard webmail clients available for customer use, but they use IMAP. With the frequency we're seeing this problem, I'd expect it's more likely to be something newer or more commonly used.
Presuming you've been able to identify which users this is affecting I would suspect you could go back to those users and determine what clients they are connecting with and then interested parties (dovecot devs?) could perform further investigation in a lab or whatever to determine what is going on. Maybe the client(s) is/are just whacky or there is a bug somewhere.
You can also track down the source IP addresses which may give you an idea as to the client as well. If it is a RIM subnet then you may be able to assume it's a blackberry. If the PTR record for the IP is webmail.somecompany.com then you can probably contact the company and discuss with them. Etc. Some companies may have a proxy or something that is attempting to hold the connections open for faster response times maybe geared for slow link connections. People do a lot of "interesting" things from time to time.
I also didn't see you mention it but presumably you are on a relatively recent version of Dovecot considering you are examining the source code.