It is not so much the warmth, but most of all the circulation of air which dries the mushroom. So, one possibility would be to build a rack and place a small electric fan under it. But of course it is better and cheaped with burning wood =)

I used to have that rack on top of my previous fireplace. Now I just placed red bricks on top of the stove and placed the rack standing on the bricks. Today, when I was at home I didn't have the rack on, I just burnt a small fire for several hours - to cook food and to warm the house. Then before I left for work the fire was out and I placed the rack on top of the stove which was still warm. This system seems to work nicely.