How do I test Millivolt valve system - Pilot is on, but main burner not lighting

Pass/fail limits for these components seem pretty wide.

If the load voltage is half that of the open circuit voltage then maximum power is transferred from source to load and this is desirable for a setup like this.

Run some experiments. Your t'pile can be simulated with a D cell and two 6.2 ohm resistors and your valve with a 3 ohm resistor.
Some t'couples only put out 28 mV but I can't find a spec on their internal resistance.

http://www.electro-tech-online.com/

 

For anything electrical to work, there must be a complete circuit from the source of power back to the power source. A thermocouple is a power source. When it is disconnected from it's load (the gas valve), and you measure it with a high impedance multimeter (this is almost like 'nothing', since it's internal resistance is generally quite high) you are measuring it as an open circuit. This is sort of like measuring the output of a D-cell when it isn't in a circuit. A closed circuit generally consists of a power source, and a load the power source is designed to run. A short circuit is where the power source is connected directly to the return on the power source, bypassing the load. A short-circuit is generally catastrophic - things overheat and can burn up.

To test the thermocouple with a load, out of its normal circuit, you'd take the two leads and connect each to one end of the resister, then measure the voltage across the resister.