I currently use gmail and a google apps account from my college, but the last one doesn't work properly, due to the lack of support for OAuth2 authentication for IMAP accounts. It's really a funny thing, since the GData plugin does indeed work fine using that same authentication (I use it with my gmail).
The company I work for changed the authentication to use MFA for Microsoft Exchange accounts. This change included that the IMAP server now supports only XOAUTH2. The client log: [17:15:17] IMAP> 1 CAPABILITY [17:15:17] IMAP< * CAPABILITY IMAP4 IMAP4rev1 AUTH=PLAIN AUTH=XOAUTH2 SASL-IR UIDPLUS MOVE ID UNSELECT CHILDREN IDLE NAMESPACE LITERAL+ [17:15:17] IMAP< 1 OK CAPABILITY completed. It looks like it says it allows plain but this seems to be wrong. So XOAUTH2 support is needed to login.
This bug has been overtaken by development in the last 3-4 years. OAUTH2 access is now included for Gmail and Microsoft Exchange. If an unusual configuration is needed this is being addressed via bug 4579. This bug could be closed.