I am having the same issue. When I open the camera app, from anywhere, it shows no picture - just black as if I have a lens cap on the camera. I can see the camera button and the flash auto/on/off, HDR text etc. If I try to reverse the camera, the grid animates away, but it remains black. It also seems the app freezes at various times and I cant switch options like the flash setting. Sometimes I can - seems random but I havent played around to determine the conditions.
My flashlight icon from bottom-swipe gesture is not greyed out, but when I press it, the icon illuminates as if it is working, but there is no LED/Flash activity.
My phone was also getting very hot and draining battery rapidly. I belive that if you turn your camera flash to off, and keep the flashlight icon OFF, that it doesnt get as hot. Not scientifically certain about this but my phone appears to have returned to normal battery performance.
I tried to let the phone drain to death and recharge, but that did not fix it.
I backed up and wiped my phone, and that did not fix it either.
I took it to the Genius bar and they ran a diagnostic, and found nothing wrong. He even opened the phone to make sure the cable was still properly seated, but that was not the issue.
My phone is 2 weeks out of warranty, so they said unfortunately nothing they can do but have me pay for a replacement or repair, but that they wouldnt recommend repair because they dont know what is wrong and it could cost more than just replacing the phone if they have to replace a bunch of things that dont end up fixing the problem.
The tech's opinion was that the LED might have been left on for a long time, and may have burned the motherboard or something.
At this point, I am just using the phone without a camera.