phone wont turn on shattered screen

The first thing to do is to totally disconnect the screen (always disconnect the battery first!). Then connect the battery back to the logic board and then connect the phone to an iTunes enabled computer to see if it is recognized. If it is, then you may only have a screen issue. If it isn’t, then you may have a logic board issue.

The heat concerns me as that is not normal, even with a busted screen. It could be caused by a bad FCAM flex or a damaged screen. However, if these variables are removed and the phone still doesn’t boot, then consider having this looked at by an experienced micro-solderer.