I have the same problem. I have a brand new iPhone 6 Plus. I upgraded from an iPhone 5 specifically to get TouchID and a larger size. Well, it IS bigger, but TouchID does not work. I can scan any finger and it will work for an hour or two. Gradually the recognition gets worse until by the next day the print is no longer recognized.
Please do not reply with Apple's canned suggestions that blame the user. The fact that I can enter a new fingerprint and it will work for awhile indicates that the lens is not dirty, my hands are not deformed, etc.
I've already gotten a replacement, but since my symptoms are the same as the ones experienced by the people bitten by the 8.0.2 bug it seems like there is still some kind of edge-case problem with 8.1.2. There are enough of us reporting in various places, that the problem should be escalated to engineering and some kind of a stopgap should be offered. How about a downgrade to before 8.0.2 for people who just want TouchID, one of the primary advertised features, on their very expensive phone to work?