Use touch ID without locking your phone

Solution 1:

This is not exactly what the OP asked for, but it might partially solve the problem:
- Enrol a fingerprint and select a passcode,
- disable "iPhone Unlock",
- change "Require Passcode" to the maximum ("After 4 hours" in my case).

This will allow you to use Touch ID in Applications, but will require passcode for unlock only once in a long time.

Solution 2:

This is not possible. You need to have a passcode to use it due to the way iOS encrypts passwords. Your iTunes store password would not be encrypted if you were not using a password, even with the Touch ID sensor.