I've just filed a bug report without logs. I've downgraded my device to 9.3.5. If there are people who is on iOS 10.0.1, please take a look at Touch ID logging section and make a log of errors. Then, post it to bugreport.apple.com.
TouchID logging profile: https://bugreport.apple.com/download/instructions/TouchID.mobileconfig
Enabling Logging
1. Download profile and install If necessary, email profile to device and install from email
2. Restart device
3. Reproduce issue
4. Sync device via iTunes
Log Locations OS X: ~/Library/Logs/CrashReporter/MobileDevice/[Your_Device_Name]/BioLog
Windows 8, 7, and Vista: C:\Users\[Your_User_Name]\AppData\Roaming\Apple Computer\Logs\CrashReporter\MobileDevice\ [Device_Name]\BioLog
Disabling Logging
1. Launch Settings
2. Tap General > Profiles
3. Tap appropriate profile
4. Tap Remove and enter passcode if asked
5. Restart device
IT IS VERY IMPORTANT TO ATTACH LOGS TO HELP FIXING THE PROBLEM.