The GPS receiver itself uses barely more power than the battery uses when sitting on a shelf. Switching the LCD display on and off probably uses more (and that's almost zero).
There is a problem with the phone - a display problem - but even leavng the GPS receiver on full-time isn't going to drain much battery.
I saw this earlier and tried this. Go to settings>privacy and safety>location>google location history>turn it off. Even though my location is off it kept pinging it. since I turned google location history off its stopped.