Supporting foam board melts when cutting plywood
Never tried this, honestly don't see a point in doing it this way (expensive material, and yes, foam melts.)

Ignoring the option of using a blade that causes less problems, a supporting material that costs less and does not melt would be appropriate. OSB, or MDF for two, or hardboard, for another. Hardboard being the generic term for Masonite® which you might or might not know it better as. And I do recall mention of using it for exactly this purpose on difficult materials in some old woodworking text. 

However, as a rule the combination of the correct (and sharp) blade and the correct depth of cut for that blade (so it's cutting into the surface, not stripping it off - this problem is usually associated with inadequate tooth projection from the cut material - the entire tooth down to the gullet should be sticking out below the sheet) are a more common solution that does not require a bunch of sacrificial backing material. A dull blade will tend to beat the wood more than it cuts it and aggravate this problem.