Is Apple going to store my fingerprint in a database if I use the iPhone 5s / Touch ID?
While Apple has provided few details regarding how they accomplish fingerprint authentication without doing so -- likely due to the fact that the technology is proprietary and protected -- Apple has stated clearly that they will not be storing anyone's fingerprints.
According to the official Touch ID video:
All fingerprint data is encrypted, and stored in the secure enclave in our new A7 chip. Here it is locked away from everything else, accessible only by the Touch ID sensor.
It's never available to other software, and it's never stored on Apple's servers or backed up to iCloud.