Help Me to Identify what is the Cause for Heating with this Circuit
If you're really feeding this 15 VAC (not 30 VAC as shown in the diagram), your input bus is around 19 VDC, which is forcing about 20 mA through R1, and causing it to dissipate about 400 mW. This is too much for a 1/4W resistor, and even a 1/2W resistor would get rather warm. Either get a bigger resistor (1W), or pick a higher value to bring the current down. 2 k&Omega; would still put about 10 mA through the LED, and the resistor would only dissipate about 200 mW.

I can't explain the droop of the output voltage without more information. How much current are you drawing at the output? Is the LM317 getting warm as well? If so, put it on a bigger heatsink.

Is the voltage at pin 2 of the LM317 also drooping? If so, perhaps R3 can't handle the current you're putting through it, although I would think that 1.25 V / 220 &Omega; = 5.7 mA should be fine for any potentiometer.