If screen is overheating you may have a short either at board level or at screen level. Try starting your phone without screen by plugging in a lightning cable with battery connected. See if it is alive and recognized by a PC. You may also try shine a bright light when the screen is in place..if you see a dimmed out image it may be a backlight problem.
Problems may present when installing the screen if battery is already plugged into the board as some circuitry is already powered and it may happen to shorten circuits during install. Unless your old battery was completely dead, try putting in the old one to rule out a bad battery. Unfortunately there’s a number of potential issues and you need to do some further testing to try and gather additional info.