Divide and conquer.
Always check power first; 5VDC and 24VAC at the correct points including the chip power pins. Although, I think you've confirmed that already.
This is usually how I approach things. Typically I try removing any items that can be removed without necessarily affecting the circuit. Upon looking at the circuit, and before disassembling the iron, I thought this would be a slam dunk repair. Once I saw a PIC, I sensed it could be trouble.
Also, on a side note, I'm fully aware this is a thermocouple, however, after many, many years of working with thermistors, I constantly call them thermistors by accident; so bare with me.
One of my temp meters (four channel) has K-type thermocouples that I figured could be used to trick the base. Upon powering it, the green LED illuminated, so I began scope probing pin 6 of the PIC, and suddenly the LED turned red. It almost seemed like each time I touched the probe to the pin, the LED would change colors, however, then it began doing so on its own. I wiggled the power wire from the transformer, and it seemed to consistently cause the LED to change so I used my (cheap) soldering iron to reflow the two power wires.
After I connected the iron which still remained cold, and that's when I probed the resistor between the base and PIC. It had 250us wide pulses occurring approximately every 8ms. I asked myself how its getting pulses, yet the iron remains cold, so I touched the iron to double check, and burned my finger.
Now things are still screwed up because if I apply power to a cold iron, the dial on minimum temp, the green LED is on (I think it should be red), and the green LED never seems to turn red or off.
Not knowing the temperature of the soldering iron (which was enough to melt solder, and burn my finger), I connected one channel of my four channel temperature meter to the iron (obviously I didn't have the best thermal conduction, but it was enough for a rough idea) and saw the iron settled to about 500 degrees F.
Regardless of where I placed the pot, the thing seemed to have a mind of its own. Sometimes it would eventually begin to cool (long after I turned down the pot), or other times it didn't begin heating immediately, but in both cases, it seems the LEDs weren't showing the correct state.
I need to somewhat start my analysis over, and check the 33uF cap. Maybe it's causing lots of noise (or one of the other caps), but I measured pin 5 of the PIC had about a 6v peak 60Hz wave that looked fairly clean.
My only other thought due to these strange occurrences: the PIC is flaky thus why it's not triggering the correct LED state and not getting correct thermocouple feedback.
Edit: reading the manual (who reads manuals nowadays?) maybe the red LED was never on. From my interpretation (and wanted to edit this quickly before someone spends time trying to analyze why the red LED isn't on) it looks like red is only for 'locked' mode.
A steady green is the temp ramping up, blinking means it has reached temp.