What a nasty! I have recently been learning to fix phones and had great success until my wifes 5c came along! I have done 5/6 5c's with no problems and got my wife a 5c to test screens with. I have had no end of problems with it and "new" screen.
From black screens but you can see the back light, three white stripes down the right hand side, white squares etc etc all with a non responsive touch ive had it all. I thought I had broken it. Even the original screen was the same when I tried to get it back on although this has been replaced at some point so was not an Apple original screen.
When I plugged just the LCD cable in the screen would be fine could see the screen lock slider on screen the lot. As soon as I plugged the digitizer cable in the above would all happen again which of course points to a problem with the Digitizer cable / plug or around that area.
There is under the black tape covering the logic board just above the digitizer socket in the top left corner a nodule coming out of the board. Not sure whats its for and not sure if it interferes with the digitizer plug seating correctly onto the logic board but have been focusing around this area. Also read the comments above ref the exposed digitizer connector and LCD connector on replacement screens.
Not sure how but I have resolved the issue after 3 days of banging my head on the desk. I fitted another screen that had been playing up as above but instead of fitting with the screen at a 90 degree angle I let it lean further forward away from the phone so I could get a real good look at the connector. Once I was sure it was connected I then got a nylon spudger and let it rest on the digitizer cable and started to bring screen back up to the 90 degree angle to fit the lcd cable and earpiece / camera cable and hey presto it worked. The spudger was then inbetween the digitizer cable and lcd cable so they were not touching. I removed the spudger and it all stilled worked and the touch screen was good to go.
Closed it up quick and tested it. The front camera would not work so did a hard reset and thats back and the screen was flickering at the top but that has settled as well.
I now working on 2 theories. The spudger has either held the digitizer connection in place whilst I have connected the LCD and Ear speaker connector or as it created a gap between the digitizer connection the LCD cable which lays on top of it it protected it from interference. It makes me think its in the area as like I said when I connected just the LCD connector and powered the phone the screen was fine.
Im going to do some more testing with the above, however it will have to wait till after the weekend. The wife wont let me go near her phone at the mo!
Will update once had another go.