Same issue. Touch Id worked FLAWLESSLY for some time (maybe 10 days, I dont remember) after I bought 5s. It worked with sweat fingers. I remember, I was on a concert, I was completely wet and it worked with no problem!!! I played with It a lot, re-recorded the prints many times and it worked!!! But than it started to fail about 75% of my attempts, sometimes its really hard even to record the print (getting the "fingerprint can not be read" errors). I think, that 7.0.2 or 7.0.3 update ruined my touch id. Now, the print only works for some time after recording. It can be hour or 15 minutes and than it starts to fail, sometimes cleaning the sensor with t-shirt helps, sometimes not... But,seriously, for the first days it worked without any cleaning...
It looks like a software problem, that came with one of the updates.
I am very disapointed, that the issue is not wide-spread... The more people will have it, the more attention it will get. So if you are reading it and have same issue, PLEASE, log in and write about it, DONT BE SILENT!
Maybe, the "As you use Touch ID, it will continue to learn and improve recognition of your fingerprint." feature was affected and started to spoil the experience with every unlock.