The solution to this is to turn off location services for the weather app and set the location manually.I made the same mistake with LI, which uses the built in iOS weather widget. I selected 'show local weather', and had forgot that it uses the GPS to fetch your location. So everytime I brought up the lockscreen, I would see the GPS arrow and hello battery drain!
Unless you're travelling to new places every day then it's not an overhead to change it when you do move.