I understand this is a VERY old topic. Nevertheless, may I post my finding here, as I feel it relevant, and documenting the technical side here could probably be interesting for future readers too. These soldering stations are still being manufactured and sold.
I recently bought a disfunctional Fahrenheit 28003 soldering station (with the same ZD-919-2 board shown above), with its belonging iron, told as the defective part. The station looked neat and clean, inside out. The iron indeed shown an open circuit. Before purchasing a new heater, I wanted to understand how this control circuit worked - this is how I got here.
I opened the faulty heater element to get an idea, what is going on with this simple two-wire iron, that has to do with some kind of reasonable temperature control. Then I found the "trick": the iron's heater element consists of the actual heater coil, and a sort of thermocouple, in series with the coil (see my photos below).
So then yes, there is a kind of temperature control: the control circuit is watching the current flowing through the heater element, via the voltage across the 0R33 resistor amplified by opamp-1, then opamp-2 compares it to the voltage set by the pot for the required iron temperature, and triggers the triac via the 555 timer. Simple as it gets.
In my case, this thermocouple was broken. I redid the thermocouple joint - just a makeshift thing to get the iron working for the time of some quick experiments. Based on my very rough measurements, the resistance of the heater element (coil+thermocouple) varied like this: at room temperature (25 °C) ~3 Ohms, then at 100/200/300/400 °C approx. 20/27/35/42 Ohms respectively - in this practical range looked actually fairly linear.
Regarding the burnt 0R33 resistor (seen this issue also on other forums): I guess it happens when that "hidden" thermocouple fails and the control circuit keeps triggering the triac.
I truly can't believe that this resistor could burn when the soldering station is switched on without the iron attached. More important though is to use it with its own iron exclusively (with this special heater element), to keep the control circuit working in its appropriate range.
(One last commment: to improve the stability of the temperature control, I added a zener diode in the positive voltage rail circuit too, by simply soldering it in parallel to the elco's leads.)