We had a toilet moved in remodeling a bathroom. We told the installer to leave room behind the toilet tank so that we could replace the paneling with drywall. He only left about 1/4 inch and so it looks like we would not be able to have room for the drywall as the paneling is only about 1/8 inch thick. Are all the tanks the same distance from the middle of the plumbing.

What you have is most likely a 12" rough toilet. You can purchase a 10" rough toilet which will give you plenty of room for your drywall. Some brands of toilets (not many) actually make a tank to convert the complete toilet to a 10". Pro-Flo comes to mind. With a Pro-Flo toilet, you can get the standard bowl, or ADA bowl for twelve inch rough, but order a ten inch rough tank.

If you loosen the floor bolts, you can usually wiggle the bowl out another 1/8" without damaging the wax ring. Then you could try 1/4" drywall or possibly 3/8".