The input to the LM317 is around 12 volts and the output set with a trimmer is 4V exactly with a load current of 1A.
I forgot to mention that there are 10 MMIC's ($450 at stake) all driven by this supply, so that's why I am a bit concerned in case the voltage exceeds 5V.
I was just giving you an idea, and of course every detail affects the design. So is the load 0.5A for 10 MMICs in parallel? And does that vary up and down? What is the absolute maximum peak load?
Regarding the zener implementation, are you advocating the use of a zener at the output of the supply in parallel with the load and a 1A fuse just directly at the output of the 317, such that the zener crowbars the fuse if the voltage of the zener is exceeded?
Fuse on the input side of the LM317, although it would work either way if you went the fuse route. And a zener is not quite the same as a crowbar.
You mentioned that "The power resistor, if selected appropriately, would both let your LM317 run cooler and allow the circuit to continue to operate properly even if the regulator fails shorted. "
I agree that a resistor would allow the 317 to run cooler , but do not understand why the circuit would continue to operate properly even if the regulator fails shorted?
How can I ensure that the voltage output does not exceed 5V if the 317 indeed gets shorted?
By selecting the appropriate resistor and zener. And that depends on what you need for a maximum current. If you can live with a maximum current of 0.5A, then you would select a 10 ohm power resistor and one of the 5W zeners I mentioned (there's a small issue here selecting these and I haven't figured out the datasheet yet). At 0.5A, there would be a 5 volt drop across the resistor, leaving 7 volts on the input of the LM317 which is adequate for a 4V output, but has a low power dissipation in the regulator of 1.5W. If the regulator goes rogue, then the zener will step in and start conducting at ~4.5V, which will result in the other 7.5 volts being dropped across the resistor and the total current to be 750mA. If the MMICs draw 0.5A, the current through the zener is 250mA and the power is ~1.2W. If the MMICs draw no current, then the zener current is 750mW and the power is ~3.6W. Toasty, but not fried if the circuit board is designed well.
If you need higher currents, you'll need to adjust those values and you might need to fuse it instead. I don't know how big a deal having the whole thing shut off would be. Or, perhaps use two 0.5A regulator circuits. But really, I think if you use fixed resistors with a large-value parallel trimmer, you have eliminated the most likely cause of an LM317 going rogue and beyond that they are pretty reliable and will shut down rather than short out.