Should I not hear a click when I push the arm of the dispenser.
Yes, if you use to hear a click before it stopped working.
Was the clicking you use to hear from the dispenser or the water valve at the back of the refrigerator?
Should I use the hair dryer on the tubes inside the refrig.
Would not hurt anything. If you have flow it's probably fine, but I have had flow on some and when pressure is added the ice plugs the line up.
At this point I would start troubleshooting with a multimeter and see if I have voltage to the water inlet valve at the back of the refrigerator when the dispenser was activated.
If you have voltage to the water valve I would first make sure it's not frozen up and if not replace the water inlet valve.
If you don't have voltage to the water inlet valve then I would check the dispenser area and the wire harness at the door hinge.
Does your dispenser have a 'lock out' feature? Is it turned off?