Nate, I am experimenting with different heights off the possum board. I'll let you know if my ears can tell a difference in tone.

When I build one with a double back, I usually only give it about 1/8th of an inch gap. 
I suspect that the size of the gap would have a subtle impact on how the sound waves interact with eachother. To some extent, the space between the bottom of the instrument and the possum board or false bottom could be considered a separate "soundbox" that will produce some amount of resonance on it's own, and most likely affects the tone in a small way. It may be useful to set your dulcimer on a couple of chunks of wood that are sitting on a board as a simulated "possum board" and see if you notice a difference in tone at different heights.
I personally have noticed that when the instrument is closer to the board, it affects the high end of the instrument tone and can make the sound more "nasally" whereas a larger distance between the dulcimer and possum board has less of this effect.

A double-back is a permanent part of the instrument, whereas a possum-board is a temporary, removable part that the instrument can be set on. 

What is the difference between a double back and a possum board?

My Galaxes have about a 1/4" spacing between backs, but you could even get by with 1/8" if you wanted to.

I agree with John. 

Ken

"The dulcimer sings a sweet song."

Thanks John. What is the spacing on a Galax double back?

Don't overthink it, Richard.  I have not run across any standard spacings myself.  I'd say a space of anywhere from 1/4" to 3/8" would be fine.  You don't need more than that.

Thinking about a possum board, Does anyone know if there are any standards for how much space between the dulcimer and the possum board?  Any science to it?

I have seen what seems to be general recommendations for 3/8:" to 1/2" spacing. 

What are your thoughts?