You shouldn't need any fuses as long as the cable and terminal posts are rated to carry the full current. The PC PSU will be internally protected so short circuits shouldn't be a problem.
I have three main concerns:
1) The power supply may need a minimum load current to give the correct output voltage.
2) The 0V rail of most PC power supplies is bonded to earth ground so the ground terminal of the 0 to 22V LM317 supply won't really be 0V but -12V which would cause a short circuit when you connect it to an oscilloscope and the LM317 will burn out, if the output voltage is set to < 0V its output is connected to real earth/ground.
3) Although you don't need any fuses to protect the power supply you might need a fuse to protect the circuit connected to it. It depends on what you're using it for, if it's just to power small MCU/logic circuits, it's prudent to have an output protected with a 1A fuse.
You should also be aware that the LM317 can't provide 1.5A from 1.2V to 22V, you'll only be guaranteed the full 1.5A when the voltage is set to between 9V and 21V. Above 21V, the voltage might drop below the set value when the load is connected and below 9V the LM317 will reduce its output current to protect itself from overheating. This is all assuming the LM317 is on a really good heat sink.