Okay, next time you go to turn it on, press and hold the power down button for a second before you press the power button, then while holding the volume down button, press the power button. If that brings you to the same screen, instead of turning the phone off by removing the battery, press and hold the volume button, then while holding that, hold the power button until it turns off. When it turns off, don't let go of the volume button, but release the power button then press and hold it again. You should boot into the fastboot menu.
I hope those directions made sense. In that menu, there should be an option to factory reset the device. That might be what you have to do. Try restarting it using this menu before you wipe the device, just to see if we can save it!
Did you notice the phone was hot? Did anything weird happen before it just shut down?