I have found additional info on this problem.
These solutions were posted by HansC and rdehavenl on this discussion:
https://discussions.apple.com/thread/7297609
-Restart the Mac in Safe mode (keep Shift button pressed after the apple sound comes on)
-Then try and open each Office 2016 application
-When that annoying Dialog comes on, Click "Always allow" (somehow in Safe mode this button works!!)
-Restart the Mac normally and voila, Office works.
Safe-mode will work, but if you know of any applications that are running to intercept user-input, try quitting those applications to allow the requests. You may need to ForceQuit and restart the office app after stopping the other apps.
Example Apps:
MagicPrefs
Hazel
AppleRemote Desktop (Remote mode)
Bomgar (Remote Support Mode)
TeamViewer (Remote Control Mode)
Linc Davis mentions changes by Apple in 10.11.1 https://discussions.apple.com/thread/7326759
First, if you're trying to enter a password remotely via a screen-sharing application such as Apple Remote Desktop or "TeamViewer," that won't work in OS X 10.11.1 or later. It was blocked as a security risk.
If you're running third-party software called "BetterTouchTool," "MagicPrefs," "Synergy," or anything else that modifies the function of an input device such as a trackpad or mouse, either remove it or refer to its developer for support.
The same thing may happen if you're using a Wacom graphics tablet.