Out of curiosity, was replacing the screen the first thing you did to try and diagnose the problem of not turning on, or was there another reason and the phone was turning on previously? How did it start; did anything change recently such as an IOS update, or any kind of damage like being dropped, getting wet, etc.?
As @tomchai mentioned, there are unfortunately a lot of different reasons why a phone doesn't turn on. Do you get any kind of indication such as a chime or vibration when you plug in a charger? You've tried different chargers and cables, right?
Go through this troubleshooting guide and see if anything there helps. If not, come on back and add the information on what you find to your question and we'll see what we can do to help.
[[iPhone Wont Turn On|iPhone Won't Turn On - iFixit|new_window=true]]
Good luck; let us know how it goes!