how to turn off passcode on iphone 5s

You can not use Touch ID without a Passcode.

iPhone 5s: Using Touch ID

Setting up Touch ID

You can set up Touch ID in iOS Setup Assistant or by tapping Settings > General > Touch ID & Passcode > Touch ID. To configure Touch ID, you must first set up a passcode. Touch ID is designed to minimize the input of your passcode, but your passcode will be needed for additional security validation, such as enrolling new fingerprints.

If you're like me you most likely found this post because you are trying to turn off ONLY the passcode and NOT the fingerprint as well on your iPhone 5s, because the title of this post is "how to turn off passcode on iphone 5s." I just though I would provide the steps to perform the task of the exact title given to the original post. The following steps using an iPhone 5s with the current iOS software available on today's date 9/23/15 which is iOS 9.0.1

Using iOS 9.0.1

#1 Tap on "Settings" (grey colored icon that looks like a gear)

#2 Tap on "Touch ID & Passcode" (scroll down a little on the "Setting's" page, red icon with what looks like a fingerprint in white in the center)

#3 Tap on "Turn Passcode Off" (scroll down on the"Touch ID & Passcode's" page it is under the grey "FINGERPRINTS" section.

#4 Now you will need to re-enter the passcode correctly to complete the process. Congratulations you're done!

settings>general>passcode lock

Tap Settings -> General -> Passcode Lock -> Enter passcode -> Turn passcode off -> Enter passcode again.

And you have also turned off Touch ID. The OP wants touch ID only, with no passcode. That is not possible.

This helped so much. Thanks. Problem solved.