Bad capacitors. It's always bad capacitors (and it's never Lupus).
They've dried out and become resistive. In the process they sometimes make life ugly for power transistors and diodes.
If you're not safe with some mains voltage (there may be some in the hot side of the supply) please don't open the box.
Look for swollen electrolytic capacitors (metal cans). Replace with reputable parts with same values (no consumer components supplier will sell parts as nasty as a cheap manufacturer can buy wholesale).
Check diodes and transistors on output side for blowage.
Main high voltage mains side capacitor might be dried out. Shows up as overload in feedback paths and shutdown. Replace it. Must be same voltage rating!
Go see the sci.electronics.repair
FAQ.
I suspect your problem is lack of connection for the sense lines. If you take a normal power supply, with just separate insulated wires in a bundle, and look at the motherboard connector, you will notice that several pins have two wires crimped together. The larger gauge wire is the actual power wire and the smaller gauge wire is the sense wire.
The purpose of the sense wire is to overcome voltage loss due to resistance in the wire harness: the power supply increases the 3.3V (for example) so that at the other end of the wires, you get true 3.3V. These may also be a sense wire on ground.
If the sense wire is not connected, the power supply detects a fault and shuts down. If you cannot tell which wires are sense in your original harness, you will need to connect together ALL +5V wires, connect together ALL +3.3V wires, connect together ALL +12V wires, and connect together all ground wires.
Edit: I just checked the ATX specification and it appears that there is probably only a sense line on the +3.3V. Try just connecting together all of the +3.3V (orange) wires.
Additional edit: the answer to the OP's specific question is no, no resistor is required, as shown by the fact that the power supply works with its original cable harness.
Best Answer
Use a NPN transistor. PS_ON to a 1k resistor at the collector, 1k resistor at the base, emitter to ground. Voltage at the base can be different than the collector.
Update: Further research shows the PS_ON tends to have a pull-up resistor to 5vsb. In that case, adding a 1k to the collector creates a resistor divider. If too high a ratio, then the input signal might not cross to a Input Voltage Low. You can size down the resistor, or omit it.