I have spent time in 5 sessions of "Chat" and "Phone" support with Apple (over 3 hours on computer and phone). Late last night I did a Restore and it worked until approx 10 am this morning. While on the chat this morning, while attempting various approaches, the Touch ID worked until I did an edit in the settings to change from "Finger 1" label to "Right Thumb" and for the other labels in the bank. Touch ID immediately quit working. I laid it down on the desk top while conversing with support and after a few minutes, I changed the labels back to the generic "Finger 1" etc. Touch ID started to work.
However, when attempting to add another finger, the sensor would not recoginze that it was being touched. At that time Touch ID would not allow opening the iPhone (from off). Again laid the iPhone down and after about 5 minutes I was able to pick up the iPhone and access via Touch ID. I immediately went to the "Add another finger..." and the sensor started to work, but quit partway through the process. From "off" condition, Touch ID would not work.
Conclusion is that temperature matters (being held warms up the iPhone and sensor, laid on the desk allows it to cool); this to me indicates a sensor, hardware issue that was exacerbated when OS 7.1 was installed.
I have an Expedited Replacement process in the works. I'm not doing anymore troubleshooting at this point. Currently Touch ID is working when used straight from the holster, for at least one fingerprint (all other fingerprints have been removed).