Sometimes a switch will short out for some reason, resulting in voltages shooting down a wire to energize a component whether the switch is energized or not. In your case, what would be suspect would be the wire leading from the "C" terminal of the ignition switch to the primer solenoid. This would cause the solenoid to be engaged constantly to a point where it would overheat and internally melt down to result in a dead short.

The thing to do is to is to see if voltage is active on that "C" terminal of the switch with the key ON (engine not running) without activating the primer solenoid portion of the switch.

If so, replace the ignition switch if you have a push key type switch that activates the primer solenoid.

Also check to see if the primer solenoid itself has failed by applying 12v to the voltage terminal of the solenoid. Does it click in and activate?... if not replace that too.

Let us know what you find.