EDIT: On reading this entire thread, it appears that some people claim no crash logs are generated, thereby casting doubt on whether SpringBoard is the culprit or some other process. It could be that the Apple Discussions user mentioned below was experiencing a different crash.
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Respectfully, the reports being collected on this thread are probably not of much use to Apple.
This appears to be a crash in SpringBoard (the app that powers the lock screen and home screen) (Update: maybe not? See edit above.) It doesn't appear to be related to whether you set up your phone as new or via a backup.
It's not that the display is turning off, but if SpringBoard crashes and doesn't relaunch itself, you have no working lock screen or home screen, which is why certain functions of the phone still work, but there is no UI. I'm guessing this crash leaves the fingerprint sensor in a confused state, it might be continuously activated, which is why it will get very hot after a while.
What would be more helpful to Apple to isolate this crash is any diagnostics logs you can grab off your phone, such as this one by a user on the Apple discussions site:
{"app_name":"SpringBoard","timestamp":"2015-10-03 18:07:22.22 -0700","app_version":"1.0","slice_uuid":"2a6bd074-7054-39ec-a5b7-d016fe13c2c3", "adam_id":0,"build_version":"50","bundleID":"com.apple.springboard","share_with_ app_devs":false,"is_first_party":true,"bug_type":"109","os_version":"iPhone OS 9.0.2 (13A452)","name":"SpringBoard"}
Incident Identifier: 523EC115-B376-478F-AA7E-CB6D6B3BF17D
CrashReporter Key: f5c671fadf7bad999f395d99cc9cd633b467e409
Hardware Model: iPhone8,1
Process: SpringBoard [58]
Path: /System/Library/CoreServices/SpringBoard.app/SpringBoard
Identifier: com.apple.springboard
Version: 50 (1.0)
Code Type: ARM-64 (Native)
Parent Process: launchd [1]
Date/Time: 2015-10-03 18:07:22.22 -0700
Launch Time: 2015-10-03 13:02:35.35 -0700
OS Version: iOS 9.0.2 (13A452)
Report Version: 10
Exception Type: EXC_BAD_ACCESS (SIGSEGV)
Exception Subtype: KERN_INVALID_ADDRESS at 0x000000010a34079a
Triggered by Thread: 0
https://discussions.apple.com/message/29051664#29051664