For those who suffer from a device that runs hot and on which the usage and standby time are both very similar/almost the same, take it to Apple. It's a hardware issue.
I've had the same thing happen twice, once on an iPhone 5 and once on an iPhone 6. Both times they did a diagnostic in store and determined there was a problem with the processor that caused it to get "stuck".
Both times I walked out with a new phone and no issues.
Also, because I see this ALL THE TIME....please make sure you "lock" or put your phone to sleep before putting in your pocket/purse. Or, as another poster suggests, set it to auto-lock after a certain period of time. I can't tell you how many people I see toss an iPhone in their purse or pocket with the display still active. Those that do this and don't have auto-lock set will notice the phone is ALWAYS on....because it is....because the user failed to press the "sleep" button.