Switching a millivolt gas valve with a low-voltage digital timer
Since the timer is powered by a 1.5v AAA, your can expect end-of-life cell voltage around 1 V. So the valve control circuit should be able to deal with input voltage range 1 - 1.5 V. Not sure that your can achieve this goal with a single MOSFET circuit; there are many power MOSFETs with low resistance at Vgs > 2.5 V, but it may be hard to find a power MOSFET with low resistance at Vgs = 1 V.

To use widely available MOSFETs with low resistance at Vgs > 2.5 V (e.g. FDN337N, IRF6201, IRL6342 etc) your can consider the following circuit:







simulate this circuit &ndash; Schematic created using CircuitLab

In this circuit, expected life of the CR2032 cell should be on the order of ten years.



Update: the working implementation, as I can see, looks like the circuit below, but without R2 and R3 (the role of these resistors is explained below): 







simulate this circuit

Inside the dashed box is my guess about the timer internals.

I feel the need to repeat the warning from the comment by transistor:


  The thermopile can't give an output if there is no flame and the valve will shut the gas off. Once you introduce a battery you have a risk of a wrong-side failure mode. e.g., MOSFET gate breakdown allows current to the valve.


That's why R2 is recommended.

Also, as I have said earlier in the comments, 1 megaohm resistor between the gate and the source (R3 on the schematic) is also recommended. If the timer output is implemented as an open collector or open drain (as I have guessed), then the MOSFET may spontaneously turn on without the signal from the timer. R3 prevents this possibility. 

Please keep in mind that the MOSFET may be turned on even with picoampere leakage current, which may happen due to change in humidity or temperature, if there is no DC path between the gate and the source. 1M resistor provides such path in any possible case.

And the final note: while FDN337N works fine with the fresh 1.5 V battery, but it may fail to work with the partially discharged battery. This may be checked by an experiment.