One particular issue that I've seen recently is when moving a database from one Exchange 2010 DAG Member to another, the Sync Issues folder starts filling up with the following Synchronization Log messages:
7:57:57 Synchronizer Version 15.0.4701
7:57:57 Synchronizing Mailbox 'User Name'
7:57:57 Synchronizing Hierarchy
7:57:58 Synchronizing server changes in folder 'Inbox'
7:57:58 Downloading from server 'Client-Access-Server'
7:57:58 1 item(s) added to offline folder
7:57:58 Error synchronizing folder
7:57:58 The client operation failed.
7:57:58 Microsoft Exchange Information Store
7:57:58 For more information on this failure, click the URL below:
Checking all the normal locations, there is nothing wrong with the synchronization, yet every 5 minutes Outlook reports an error synchronizing. The above example is with Outlook 2013 on Exchange 2010 SP3 RU6, but I've seen it through several Service Packs and Rollups. The issue also shows up on Outlook 2010.
Looking through all the settings on the Exchange DAG Members, and the Client Access Server in use (which doesn't change during the failover), there is nothing out of the ordinary that might keep it from syncing.
If I find a solution to this particular issue, I'll post it back here for posterity. Until then, this is merely another "Microsoft, please fix you're sync problems" blog.