As of Oct 9, 2016 I am having what sounds like the identical problem.
I have an iPhone 6- it's actually a "new" phone from Apple (previous GPS hardware issue- replaced via Apple Care).
Updated to iOS10.0.2 (14A456).
Problem started yesterday just randomly- noted that at first camera took a bit of time to turn on, then took out of focus pics. I also initially noted that swiping to video or slo-mo activated the camera but as of today that is not working for me.
In addition, I intermittently get the yellow caution symbol at the "flash" site stating that the Flash is Disabled- Camera needs to cool down.
I have forced closed camera app, turned off/on the phone, and done the soft reset- haven't done the factory reset yet. Would rather not since others have not found it helpful and seems like a time-waster.
I have not dropped/traumatized the phone.
The front-facing camera is fine.
Looking forward to hearing about any solutions from users or Apple. I likely will be heading into a friendly neighbourhood Apple store later this week though.