What is the gas valve section of this dryer schematic doing?
When the flame sensor is closed the current flows through the ignitor and the gas valve is shorted. When the flame sensor is open, the gas valve is in series with the ignitor so the current through the latter is much less. 

Not sure exactly how you get from the initial state to the running state- with the electronic types the HSI ignitor is heated then the gas valve tries a few times to start the flame and if it does not show up on the flame sensor it gives up. Perhaps the HSI heats the flame sensor. 

Edit: according to the patent USA 380638 referenced by @joeforker, the conjecture is correct- the SiC HSI (hot surface igniter) heats the flame sensor, which is a radiant heat sensor. 

By the way, you can easily explode your house into a pile of kindling fooling with this sort of stuff if you don't know what you are doing.