Filling your gasoline tank doesn't save any gasoline in the vehicle as your vehicle will use the same amount of gasoline no matter how full your gas tank is or how low your gas tank is.
However keeping your gasoline tank on your vehicle does help extend the life of your gasoline pump in the gas tank as the gasoline helps to keep the gas pump in the gas tank cool.
If you constantly run the gas tank on low then it can cause overheating of the gas pump which leads to failure of the gas pump sooner than normal.
Most gas pumps will last as long as the vehicle however some gas pumps may only last 100,000 to 200,000 miles if you don't keep the gas tank full.
Keeping the gas tank full also helps prevent moisture build up in the gas tank and keeps you from running out of gasoline.
I always try to keep my gas tank full but if not I always keep it at least half full and if possible I refill the gas tank after it gets to 1/2 tank of gasoline.
But you won't save any gas by keeping the gasoline tank full.
If your vehicle is using more gas than it should then it likely needs a tuneup because it could have bad ignition coils, bad spark plugs, bad spark plug wires, bad fuel injectors or dirty or clogged fuel injectors as well.