...it's about 1/8" thick Masonite (very dense MDF) that is gloss white on one side; it's about $12 for a 4x8 sheet at Lowes.

That's exactly the tile board I mentioned above. Makes a great bench top. I applied it to plywood using contact cement, just as you would apply formica. I don't think it's quite as tough as formica ("high pressure laminate"), so I have a few gouges and stains accumulated over 20+ years.

 

Using glass, a piece of copper foil kept in close contact with it, would it help in reducing the collection of static charges?

I do not mean covering the whole thing, just applied on one corner and duly grounded (earthed).

 

Glass plate in contact with other surfaces (like a wood table top) doesn't get ESD issues. We're not talking about rubbing a little glass rod on a silk rag.

You should be wearing a grounded wrist strap and will be constantly touching the glass plate, and the metal parts of the thing you are soldering are touching the glass plate, connecting through to the grounded soldering iron tip.

 

My MDF bench top needed repainting, I saw some plastic self-adhesive floor tiles, tougher, quicker and easier and half the price of paint; not sure that using them was a good idea ESD wise.