Issue:
Managed metadata SharePoint document properties can not be edited in the rich client Office applications starting in build 16.0.7167.2040, this is a defect and not by design.
Screen shot of working (prior to 7167)
Not working (7167+)
Workarounds:
Roll back the Office 2016 client to a version prior to 7167 or edit the properties with the SharePoint UI.
Current Status:
The product group is actively working on this issue right now, check back to this blog post for updates.