Something similar happened to me. I replaced a broken screen with a new one from iFixit.com and it didn't turn on. I put the old screen back and tried to turn it on, and that didn't work either.
Also, connected to laptop to see if I could see it getting charged. iTunes showed a message that the screen was locked. It made me think that perhaps the phone is on (which it wasn't before replacing the screen--I had turned it off), but there was nothing on the screen.
Then I thought of reseting the phone using the new screen (press and hold power and home button together for approximately 10 seconds). Then I turned it back on and it worked! That's all!
I hope no other part is malfunctioning in your case.