Found this in an old note file I had to help a friend with a similar issue.
There are a number of things that can be wrong, starting with software corruption AND/or hardware failure.
What everybody here is saying: you must first try to restore from DFU.
1/ Disconnect the front camera, leave it off.
2/ Connect a known good battery. Known good, not necessarily new.
3/ Connect screen (LCD + touch connectors only).
4/ Connect home button.
5/ Connect to computer. Turn on iTunes.
– Restore from DFU:
6/ Press and hold Power Button for 2 seconds.
7/ Keep holding Power Button and now press and hold Home Button.
8/ Keep holding both buttons for 15 seconds.
9/ Release Power Button only.
10/ Keep pressing Home Button for another 12-15 seconds.
Do you see a black screen on the iPhone and on iTunes the message “iPhone in restore mode detected?” If yes, click on restore.