There are two wires Grey/Red (operates low fuel warning light) and a Light Blue/Yellow (operates fuel gauge) on the fuel tank sender.
The gauge works with the ignition on and you dont need the engine running.
To check the circuit..
First check make sure that the fuel tank is earthed, if there is no earth the gauge or the fuel warning light will not work!
You will need a long length of cable (test cable) to carry out these tests, connect the test cable to the battery earth connection.
Remove the two cables from the fuel sender.
With the ignition on and the two cables removed from the sender unit, connect the other end of the test earth cable to the Grey/Red cable that you have removed from the fuel tank sender, the fuel warning light should now come on.
Try the same test for the other cable Light Blue/Yellow, this time the fuel gauge should go to full.
Report back your findings.