I just posted a similar issue with my new 5s/32 today and the fix Apple Care gave me the first time was to restore the iPhone to factory settings.
Then re-enter the fingerprints along with everything else that has to be re-entered (big headache).
A restore from backup through iTunes will not help one bit and I do like the fingerprint access to the iPhone instead of punching in the passcode (which is required after a cold shut down at any circumstance).
Hence I almost always leave the iPhone on 24 hours a day and keep charging it up every two or three days (Battery Doctor on the iTunes app store is a good one to have and has great battery tips on it).
MY FIX .. or really a work around so far.. is to re-enter in two more fingerprints into the #3 and #4 slots ... as the #1 and #2 fingerprints are still in there (Settings > Touch ID & Passcode) but are not recognized for use.
See how long that lasts (seems to lose the fingerprints every 5 weeks).
I am hoping that the problem is with iTunes as I updated an app last night and this morning lost the fingerprint data.
I would further recommend that you TURN OFF in the Touch ID & Passcode, the iTunes authorization for the fingerprint scan.
Just keep the authorization of the fingerprint to unlock your iPhone.
That might be the problem ... is iTunes in making the fingerprint lose function.
I hope so anyway.
If it was iOS software that would be more global in effecting millions of iPhone owners and I can not technically sort out how data is lost because of a hardware problem. The iPhone tells you when the fingerprint is in the iOS memory after some effort to get it in there.
Apple Care wants me to take the iPhone into the Apple Store for the Genius to make an examination but that is booked up 4 days in advance in the nearest Apple Store to me in Silicon Valley.
With almost 200 Apple stores world wide and if every one of them has 4 days of customers looking to have the Genius work their iPhone that is 800 man days of back log times 8 iPhones a day on the low side .. or 6,000 iPhones needing a look see. Not bad considering there are 50 million iPhones out there. Still it is a headache.
Apparently they only have one iMac working the Genius to take care of the customers in line at this one store and they will not Genius the diagnostics over the internet though my USB port.
I live 2 hours away (@ 70 mph highway speed) from the Apple Store so that is hard to make happen right now in matching our calendars for me to take one whole day to run diagnositcs.