Did you try putting the old screen back on the device? It is possible the new screen is drawing too much power and messing the power-regulation that the iPhone traditionally employs with the components. I've seen some low-quality or defective screens cause power fluctuations in the device in the past that was only resolved by replacing said screen.
It is also possible that the device is experiencing Tristar disease. Repair shops have been inundated with these issues as of late. It's caused by using non-Certified charging cables and accessories that power through the lightning port. It's pretty hard to tell without a full-on teardown of the cable (that would break it, obviously) but a good rule of thumb is if the cable cost less than $5 and came from a gas station, it's not a certified cable. Certified Cables, or mFI cables, employ a U2 chip that regulates power like a surge protector. If this is the case the phone will need board level repair on that chip to be up and running again.