**4/17/2017 UPDATE this issue will be resolved in the May 2017 update to Visio Click To Run 2016.
SharePoint Online opens Visio 2016 documents as read only from document library's. Visio handles read only drawings poorly and users are unable to save any changes back to the server. Users must save to a local directory then upload the file back to SharePoint.
This is a software defect.
Open the Visio 2016 files in the web viewer first, then from the web viewer pick to open the file in the rich Visio client.
Use the Visio 2016 rich client app to open the files directly with File / Open, or from the recent documents list.
The Visio product group hopes to have this issue addressed by July 2017.