I replaced a screen on a 5c and then had the same "heating" issue and battery drain. I suspected it was a short somewhere, so I dismantled the iPhone again. On doing so I noticed that the three connectors on the new LCD didn't have the small foam pads on them as per the original LCD. So I transferred the foam pads from the original screen onto the new one and re-assembled. The iPhone is now fine. I assume the foam pads are some sort of spacer to prevent the three connectors from touching.