I have the same issue.
I have a 128GB iPhone 6 (not a 6s) which was working without any issues until the operating system upgrade to iOS10. After the upgrade to iOS10, I have experienced the same issues reflected in the first post.
When using the phone, the battery percentage indicates a rapidly draining battery (the displayed percentage drops quickly) and shuts down when the displayed percentage indicates the software believes the battery has been depleted. Plugging the phone into a charger immediately results in the phone powering back up and the displayed battery percentage shows a healthy amount of charge remaining (~45+%, it varies). In these instances, I am also able to restart the phone by holding down the home and power buttons simultaneously (and the displayed battery percentage reveals ~45%+).
Needless to say, I've tried everything. Aggressively kill applications in the background, turn off all widgets, restrict location services, etc. Nothing has worked.
I was hoping 10.2 would solve this problem. It hasn't made any difference.
As a heads up to anyone thinking of taking their phone into the store:
I made an appointment at the Genius Bar to diagnose this issue. They diagnosed the battery as being healthy (well into the green area on a box diagram they showed me). Unfortunately, their only resolution for this issue is to replace the battery (and charge me for it because the phone is over a year old). However, the battery, as diagnosed by Apple with their own diagnostic tools, is fine (actually, more than fine).
The genius told me that the best thing to do at this point is to reset my phone (and install everything from scratch, not my backup) and see what happens. I'm sincerely hoping this solves my problem, but I'm not optimistic.
Needless to say, I, like many people on this post, am pretty frustrated at this point. It's extremely frustrating to have your phone die randomly, or maybe worse, when you are trying to take pictures or videos. It's also extremely frustrating to waste your time scheduling a Genius Bar appointment days in advance and taking time out of your day to go to the appointment when Apple doesn't seem to have a solution to this problem (or even admit there is a problem).
Like many people in this discussion thread, I have been a reasonably loyal Apple customer for about 17 years (many, many devices and computers), a large part of this loyalty was based on my experience that Apple worked quickly to resolve issues that arose with their products. In the past, the support I had received at the Genius Bar had been very good (sometimes exceptional); Apple made me feel valued as a customer. My recent experiences with my phone and my last Genius Bar experience have left me wondering whether Apple is taking a less supportive view of customer service.
My questions to Apple are:
1) Is replacing the battery (and charging the customer around $90 for it) a sensible resolution to this issue when the battery holds a charge and Apple's diagnostic tools indicate that there are no issues with the battery?
2) Is Apple exploring the possibility that changes introduced in iOS10 have introduced a significant flaw where the battery charge is misread (and, as a result, the phone shuts down prematurely)?
3) At the time of my posting there were ~730 posts and ~60k views of this discussion thread. Is there a system at Apple, where the most active threads on the support forum help surface bugs for the software engineers to address?