Does changing wire gauge change output of a thermocouple?
Thicker wires mean you are heating the 'cold' end more (and, as in the comment) drawing more heat from the hot end. To a rough approximation the no-load voltage is proportional to the temperature difference between the ends. 

A long thick wire (long enough the cold end does not heat appreciably) should not be any different from a long thin wire pair if the alloys are similar and negligible current is drawn.  

Think of a heat engine, if that makes sense, when you start drawing current the resistance figures into it as well as the heat flow down the wires. 

To get a good reading you have to have both ends at controlled temperatures. For a rough measurement (maybe within 2-3 degrees C) you can put one end (the junction) in boiling water and the other (transition junction(s) to copper on both sides) into an ice-water slurry.