iOS 7: delete a location from the weather application
It's a common interaction on iOS : Swipe to the left. 

Swipe the location to the left, the delete button will appear. 

But this does not work on the one that is at the top of the screen if your letting it use your current location

Swipe the location to the left IN LIST MODE. Then a red square delete button appears on the right edge. Tap it and the location is gone.