I used to make veneered lampshades and lighting fixture panels and never had any problems with finish odors. I did have some problems with the veneer cracking from the heat of the light bulb(s). I would suggest a couple controlled experiments using the species of wood you intend to use for the finished product. Subject them to heat from a 60 or 75 w lightbulb at 3"-4" distance for a couple hours and see if you have any adverse wood behavior going on. It might pay to secure the perimeter of your panels however you plan to do it, so that they are constrained.
On another note, I found that veneer is pretty delicate stuff, and that it tends to warp, crack and twist, and that some species are much worse than others. I had my best luck with diffuse porous woods such as birdseye maple, and worse luck with brittle exotics and ring porous species such as oak.
I also found that there is an optimum veneer thickness that yields the most attractive glowing effect for a given species. If you are laminating two thicknesses of veneer, you're probably already over the optimal thickness; you b the judge of that. What I did was to laminate a single thickness of veneer to a piece of clear plexiglas and then finish sand it over a light source so I could watch the glow develop as I sanded material away. I considered how it would be done in production, and my best guess was to use a Timesaver-type thicknessing sander that could be tuned to remove just a couple thousandths of material at a time. Suffice it to say that I never was able to follow through with that plan, but you might want to experiment with the plastic approach. As I recall, I had to use a very stinky and probably toxic three-part adhesive to glue the wood to the plexi, but maybe there's a friendlier adhesive around nowadays.
Oh well, hope this helps. I don't think you'll have trouble with finishes outgassing, although I'd be careful about any that are thermoplastic such as shellac. I used Waterlox, which is a simple drying varnish, but spray finishes that catalyze or polymerize would probably be the best, though they will not penetrate the wood. You might want to experiment with penetrating finishes, too. I've seen what immersion in oil finish does to Norfolk Island pine vessels. It's pretty dramatic.
You're into a fun realm here. Let us know how you make out. And post some pix of your experiments.
Ellis Walentine, Host