[e.g. Outlook 2016 16.0.4456.1003, 64 bit on Windows 10 Pro x64 v 1607]
Subject says it all - when I change the Office Background or Office Theme, and then restart Outlook/Excel/... it reverts to No Background/Colorful Theme.
I had same problem. I solved it by checking my Office Account (example using Outlook: File > Office Account). There in the User Account Info there was a message saying there was a problem with my Microsoft Account and offered to fix it. I clicked on the Fix It message and it solved the problem.
Ticking this option may help:
File > Options > General. "Always use these values regardless of sign in to Office.