Same issue for me - Outlook f2016 for Mac connecting to my O365 E3 Business account constantly prompting for credentials. I was an early adopter of Office 2016 for Mac and never experienced an issue until mid-Dec 2016. I cannot think of anything on the Mac
that changed - no password changed and no updates to OSx. One evening both my iMac and MacBook started prompting for credentials within an hour of one another. If I re-entered my O365 password approx 8x (being prompted each time) it would finally take and
work for about an hour and then start prompting again. Once it started prompting, I would not be able to reconnect it for about 30 minutes - after 30 minutes - rinse-n-repeat.
As a troubleshooting technique, I tried changing my O365 password, but that did not work. OWA did not experience the issue nor did my iPhone running Outlook for iOS.
I am running Sierra 10.12.2 and Outlook 2016 for Mac v15.31 on both machines (it may have started on a previous subversion of OL in mid-Dec but OL updated and the problem persisted).
I finally got around to contacting O365 support and they focused on whether or not I had multi-factor auth enabled (I did not), but they looked hard at it. They also focused on my Azure account. In the end, the fix was my Keychain. They said, they performed
a repair/analysis on my mailbox, but found no inconsistencies.
The following finally corrected my issue:
1)Closed Outlook on both my Macs
2)Opened Keychain Access on Mac #1
3)Selected 'Login' (top left pane) and selected 'All items' under Category (bottom left pane)
4)Deleted EVERY entry that referenced Microsoft, Office 365 and Exchange (I left the OneDrive entries)
5)Closed Keychain Access
6)Rebooted Mac #1
7)Opened Keychain Access on Mac #2 and ensured that the deletions replicated to Mac #2 and were removed. If any exist, deleted them.
8)Close Keychain Access on Mac #2
(Remember, Keychain access can be set to replicate between Mac devices. Yes, I know I could have probably deleted them from iCloud, but I wanted to ensure they were off the device)
9)After the reboot, logged to Mac #1, opened KeyChain Access and ensured that no MS/O365 related entries replicated back
10)Opened Outlook on Mac #1 and entered credentials.
This worked for me! I then opened Outlook on Mac #2, and entered my credentials and all was good on both Macs
PS: O365 Support had me delete my Outlook profile on Mac #1 before starting step #1 above, but I do not believe it was necessary because I did NOT delete my profile on Mac #2 and it worked fine.
Good luck to all because this was very frustrating.