I thought this might be because of something going on in the geocoding toolbar, and I think I have narrowed it down to two choices when right-clicking the address on the geocoding toolbar after entering the address.
The Add as Stop to Find Route and Add as Barrier to Find Route require signing in to ArcGIS Online to use routing. I'm not sure why this particular map is wanting me to sign on and others using the same geocoding toolbar don't.
I checked my extensions under the Customize menu and I had Network Analyst checked, but I have since unchecked it and I still get the annoying warning message.
I have also removed the two offending choices from the Address Find context menu, but they still show up in the geocoding context menu and the Find dialog context menu.
Jeff Ward
Summit County, Utah