Reading T/C voltage when Heater is present in series
You won't likely be able to get a good reading with the heater on, so that means turning the heater off long enough to get a reading, then on again, which means you have to have a circuit that settles very fast and recovers from an overload very fast. If it's a K type thermocouple it will have maybe 10-15mV output at soldering temperature (and a tempco of +42uV/K roughly). If you ground the end of the heater that's connected to the thermocouple, you'll have a maximum signal of maybe +/-20mV wrt ground, and with the heater on it might go to a few hundred mV. 

So, an op-amp circuit shouldn't be too difficult to deal with that, though it will have recover from overload fairly quickly. You could "blank" op-amp input by shunting it with an analog switch, which would make it settling time rather than overload recovery time. Use a low bias current op-amp and series resistance to protect the inputs. 

The disadvantage of this is that the circuit will be sensitive to noise, because you won't be able to add an aggressive low-pass filter, but it should be do-able in a controlled situation like this one. Cold junction compensation requires you to know the temperature at the point where the thermocouple materials transition to copper. 



Edit: If you have only two connections available you could do something like this: 







simulate this circuit &ndash; Schematic created using CircuitLab

In practice the half-bridge is going to need a bit more complex drive circuit to deal with 24V, but this is the concept. Getting accurate cold junction compensation will be almost impossible in this configuration unless the wires to the handset are actually dissimilar metals matching the thermocouple uV/K.