What privacy setting must be enabled to use Night Shift's Sunset to Sunrise option?
Update: Rebooting isn't required! See response below. The rest of this answer is correct.

Rebooting didn't help and I couldn't get Apple's official instructions (https://support.apple.com/kb/PH26239?locale=en_US) to work, either.

I ended up unchecking "Setting Time Zone", rebooting, and then rechecking "Setting Time Zone" after the reboot. Selecting Sunrise to Sunset was then accepted without complaint. 

Here are the complete instructions, using variations on Apple's official instructions:

Uncheck the Setting Time Zone option: Choose Apple menu > System Preferences, click Security &amp; Privacy, then click Privacy. Select Enable Location Services, scroll down to System Services, click Details, deselect Setting Time Zone, then click Done.

Optionally, reboot (Although I'm not sure this step is necessary, it's what I did. [Users have reported this is not required])

Check the Setting Time Zone option: Choose Apple menu > System Preferences, click Security &amp; Privacy, then click Privacy. Select Enable Location Services, scroll down to System Services, click Details, select Setting Time Zone, then click Done.