How to amplify a thermocouple that shares a wire with a heating element (soldering iron?)
Nichrome to copper has a Seebeck coefficent of around +18.5 uV/K and the heater  may be rather hot relative to the soldering iron tip, so I don't think it's a particularly good idea to include the heater in the thermocouple measurement circuit, though it would probably still work similarly just worse. Follow whatever the original designers of the tip did, would be my suggestion.
If the heater current is zero there will not be any voltage drop across the C1 wire during the measurement phase. You do  have to ensure the measurement circuit recovers from overload (presumably diode-limited) and settles quickly enough for an accurate measurement before the heater has to be turned on again.