I fixed this after upgrading to Windows 10 just a few minutes ago. I enabled the shortcut flag for "Run as Administrator". If it's on your taskbar, right click the shortcut and then when the small popup window shows, right click on the "Final Fantasy XIV" icon at the top of the list and choose "properties".

In the Compatibility tab mark the "Run this program as Administrator" checkbox and hit "OK".

If you want you can right-click the shortcut each time and choose "Run as Administrator", but that gets tedious.