When you fasten the seat belt power is sent to two components. One is the instrument panel and the other is the start interlock relay. The start interlock relay is in the engine compartment fuse panel. It is the largest relay in there and has 6 terminals in pairs of two. The relay could be stuck in the latched position which would send power directly to the start relay from the seat belt.
The start interlock relay normally is energized when the key is turned to the start position. Power goes directly from the key switch to the start interlock relay causing the relay to latch. It is possible for the key switch to be faulty. Since the engine starts the fuel shut off solenoid is getting power. On startup power comes from the key switch and then switches to power from the instrument panel to keep the engine running.
I would start at the ignition switch. Using a test light or volt/ohm meter check that you only have power to the "bat" terminal with the key off. If you have power at other terminals the switch is either not off or the switch is defective.
If the ignition switch checks out ok, then the start interlock relay is probably defective. Remove the relay and check for power at the start interlock relay base terminals. There should be no power at any of the terminals with the key off. With the key in the on position there should be power at one of the terminals. With the seat belt fastened, and someone sitting in the seat, there should be power at another terminal. Turning the key to the start position should power up a third terminal.`
That's a starting point. If you do find the problem, list the results of your tests and we will go from there.
