Why does a thermocouple circuit need to have minimal current flowing through it?
Thermocouples generate a small signal voltage on the order of 50 μV/°C. (Notice that it's micro volts.)  The thermocouple wires aren't ideal conductors, so they have resistance.  If you have current flowing through the thermocouple, it will cause a voltage drop due to resistance in the thermocouple wires.  This voltage drop will appear on top of the thermocouple signal, and it will be indistinguishable from the thermocouple signal.