How do I use the plastic backing on patches?
The Rema Tip-Top patches that I use--which sound similar to those which the questioner describes--come with instructions in each plastic box.  The package that I have shows the instructions pictorally in seven steps.  

Here is the product that I use:  http://www.artscyclery.com/descpage-79002.html

In step 7, the manufacturer distinctly recommends that the user remove the plastic backing.  

I usually remove this cellophane backing myself.  It sometimes is a little finnicky--if your patch isn't totally set it will seem to pull up the patch.  Just use your thumb to set the patch more carefully.  In the package's Step 6, the manufacturer uses a totally unidentifiable rectagular object to smooth out the patch before removing the cellophane.  I use my thumb, personally, and it works just fine.

If one side of the cellophane won't seem to come off without removing the patch, you didn't put down enough vulcanizing fluid.  Try the other side.  If the patch is reasonably centered, this shouldn't be a problem.

The answer about leaching chemicals is news to me--though I admit freely that I never looked farther than the instructions in the box for advice.  

Once in a moment of frustration, I left the cellophane on and put it into the tire--no problems so far.  I had some concern there would be a problem if the cellophane were stuck between the rim and the bead but this doesn't seem to have been a problem for Anthony K or Darren Cope.  

It looks as though it doesn't matter, but the manufacturer of Rema patches recommends removing it.  I say remove it carefully.  This is best done if you are generous with the vulcanizing fluid and careful to press the patch into place.