How do I keep wood from rotting in back of succulent "living art"?

You could try using a marine-grade varnish or sealer to prevent moisture damage to the wood. Keep in mind that succulents will not need to be watered very often, so that will help as well.

You can put a heavy mil black plastic over the wood, I use 7 mil contractor bags from the hardware store.

Make a Living Succulent Picture | Succulent wall - Pinterest

https://www.pinterest.com/pin/277323289524400116/

It probably will rot eventually. You might try a wood preservative (sorry can't remember the name of it, but it is a well known product), or, how about coating with a vinyl spray before the planting.

Oh, just thought of another possible solution...Flex Seal.

http://www.hometalk.com/diy/grow/succulents/diy-vertical-succulent-planter-172547

I was thinking of some of those solutions ( great minds...!) but didn't know if they would work. Thanks to you all.

Yes, you are correct; rot will eventually occur. I use Saran wrap to line anything that I know will eventually rot and it works just fine. Try and keep your eye open when disposing of items such as empty cling wrap, or foil, boxes as they might fit in the frame. You can then line the empty cardboard box, assuming it fits. I would probably use two layers of cling wrap. I've been using good quality cling wrap for ages in various baskets of mine and no rot has occurred.