Same issue here. With 10581 I had 2 System apps that would fail with error 0x803F8006 - Glance and Tips& Tricks.
Now that I'm upgraded to 10586, I've hard reset my AT&T 925 3 times already, trying difference sequences of skipping restoring backups, skipping signing in, deleting all the AT&T crud, then enabling Wifi, connecting my MS account and trying updating ~50+ items one at a time. Now all Lumia system apps, plus the HERE apps, camera ALL error out with 0x803F8006. Some update, like Word, Powerpoint, People, Mail, Maps but its not a good experience.
All I can guess is the apps and / or store is not synced up with version dependencies, etc. I'm really just grasping at ideas though.
Maybe it will all just fix itself by the time I wake up tomorrow.