Folks having trouble with their Personal Hotspot feature in iOS 7.1 aren't alone. A number of users have been complaining of tethering issues since installing the latest iOS update, as evidenced by this 16-page thread in Apple's support forums.
Specifically, the problem seems to occur when trying to save custom APN settings in a device's Cellular menu. Apparently you can enter information into the text fields, but the values don't save and any attempt to make a connection errors out...