A fuel gauge essentially consists of two windings. One winding (I'll call it the "primary") is independent of the sender unit and it generates a constant force that tends to push the needle to "full". The other coil (I'll call it the "secondary") has the sender unit (and therefore its resistance) in the circuit. When the tank is full, the sender unit has full resistance and little current flows through the secondary coil, allowing the "primary" coil to exert its force and deflect the needle to "F". When the tank is empty, the sender unit is essentially at zero resistance, allowing more current to flow through the secondary coil and it opposes the force exerted by the primary coil, pushing the needle to "E". With all this in mind, the needle could be at "F" because:
-the tank is full.
-the tank is not grounded properly.
-there is a bad connection/open circuit in the wiring going to the sender unit.
-the "secondary coil is open.
For a burned out gauge, I don't know how to fix it. The wire used for the windings is like the diameter of a hair. If you are lucky, the wiring burned out near a terminal and you might be able to resolder.
