When a lot of folks think of fingerprint scanning technology they often assume there's a single way to do it, but nothing could be further from the truth. There are actually over a half dozen different technologies -- and combinations thereof -- that various devices employ to read prints, with varying levels of reliability, and yes, some of them would indeed work with a finger you chopped off of a dear friend, but the Touch ID sensor on the iPhone 5s isn't one of them.
Capacitive - A capacitive sensor is activated by the slight electrical charge running through your skin. We all have a small amount of electrical current running through our bodies, and capacitive technology utilizes that to sense touch. This is also the same technology used in the iPhone's touchscreen to detect input.
Radio frequency - RF waves do not respond to the dead layer of skin on the outside of your finger -- the part that might be chapped or too dry to be read with much accuracy -- and instead reads only the living tissue underneath. This produces an extremely precise image of your print, and ensures that a severed finger is completely useless.
This means that the Touch ID sensor should be remarkably accurate for living creatures, but it also means that only a finger attached to a beating heart will be able to unlock it. So, should someone run up to you, hack off your finger, grab your iPhone and attempt to unlock it, there's virtually no chance it's going to work.
Once the tissue is dead -- which, in the case of someone chopping your finger off without your consent, should happen within a matter of minutes -- two things will happen: First, the finger will lose all electrical charge and will fail to even activate the sensor, and secondly, if by some chance the sensor could be artificially activated, the RF reader that is searching for a print will find no living tissue and fail, leaving the device locked.
It's important to note that in order to utilize Touch ID you must also set up a passcode, which acts as a back-up method to unlock your device. If someone really wanted to break into your device, chances are they'd be able to obtain your passcode more easily than actually slicing off a finger.
However, if by some miracle the person snatching your finger had a compatible human host waiting for your finger to be transplanted onto their body -- and if they managed to complete the procedure before the tissue died -- you might have cause for concern. Oh wait, that's utterly insane, so no, you have nothing to worry about.