I have an idea how such a transition could go:
- Upgrade the old dovecot 1.2.15 to 2.1.whateveritis from debian squeeze backports.
Backports version is a bit outdated so you might want to consider using version from jessie, which seems to follow the upstream more closely.
The tradeoff is of course the need to pull some system libraries with it as well, namely libc. Newer version may interfere with whatever else you run on the server and you'll probably need to solve some dependencies manually.
It's working for me on pure dovecot server, YMMV
Other than that I more or less aggree with the post from Terry Barnum - stop postfix to prevent delivery of new mail, block the old IMAP/POP3 server from regular users, use some sort of IMAP sync (either imapsync or - IMO - even better doveadm sync) to download data from the old server to the new one. After that's done, switch IP addresses and enable services on new server.
It'll need some late-evening/night time work and maybe console access if the IP switch goes wrong, but otherwise it should be hassle-free - particulary no need to change ppl's mail client settings this way