So I think I understand...
My existing wires look like 2 strand speaker wire. When the wire ends are touched together, it closes the circuit and the fireplace starts.
If I understand the diagrams correctly (assuming I can find a convenient place to plug in the i/o link), I would slip one wire in the n/c or n/o port (likely n/o, so it would show as "ON" when triggered the other way and "Close" the circuit on demand?), and the other in the common port. The series vs parallel is a little over my head (not an electrical guy), but I'm sure I can google it and figure it out.
Does that sound about right? Also- thanks for the help, I really appreciate the hand.