Wait for an update.
Steve - I have an iphone 7 as does my wife, both brand new as of this past weekend. Both of ours had the same issue you are referring to. Both would happen after taking a short 8-9 second video, then the flash error popped up "iphone needs to cool down". I spoke with Apple on this today, and we reset my phone to factory. Went through setup and set the device as a new device. Nothing was added or installed, and when setting up, I went through and turned nothing on (ie Siri, Location services, fingerprint, etc.). First thing when it started I opened the camera app, took an 8 second video with no flash, then the "flash disabled" error showed immediately when I went to take a pic. Clearly not a buggy app or something, and all settings were factory.
Apple wanted me to take a step further and do a re-install of the software to rule that out, but I didn't have time. I grabbed the old trusty 6s Plus I have had, and checked that camera (never has given me issue before). I can report that I took an 8 second video, and bam, "Flash disabled: iphone needs to cool down" error showed on that phone. I recently updated the 6s Plus phone to 10.0.2, as both the new iphone 7's were updated this weekend when I bought them and connected to itunes on my mac.
I know this doesn't answer the issue, but I am betting it is a bug in the 10.0.2 update, as I can tell you having taken over 130 videos and 950 pics with the 6s and 6s Plus we both had prior to the 10.0.2, this never happened.
Oddly, even with the error showing, I took a picture with the flash (3 of them) and the flash worked, and the error went away after a few moments.
