Well this happened to me when I was repairing a customer's phone. I have done about 10 iphone 5s and countless 4s and 3s. I am a repair guy.
What happened ... first the phone went together and turned on fine after replacing the screen but when I tested it the proximity sensor wasn't working. I opened it back up and fixed this but noticed when I put it back the lcd connector on the new sreen had a slight curve to it. I put it on and the proximity sensor worked but the screen was flickering so I tried connecting it and reconnecting it but it still flickered. I tried a soft reset (holding the sleep/power button and the home button till the phone shuts down, continuing holding button, till apple sign comes on then release both buttons). It then went to DFU mode, when restarting the phone apple logo comes on and at the top of the screen you will see some text at the top of the phone, and then the dredded red screen. I put on a new screen and it still was stuck tried several resets, plugged into to itunes no luck. Tried checking connection all was good. @#$%% to say the least. Well I read it may be the proximity sensor but I feel what caused this was the fact that I was plugging in the lcd while it was on. I should have powered down the phone first.
I did fix it though finally and here is how. Let the battery run all the way down then gave it a good charge. Note it still turned on in DFU mode though. Then I plugged it in to my computer and while plugged in did the power reset mentioned above. So try this fix.
1. plug in to computer and open itunes.
2. hold power/sleep button and home button down simultaneously unitll shuts down and apple logo reapears. Release both buttons.
3. let it retart. I tunes should recognize the phone and it should be out of DFU mode.
NOTE I did not need to restore the phone. Luckily.
Tip: Back up your phone to icloud or itunes frequently. :) and repair guys power off your phone before you do repairs or reconnect any screens.