I googled all sorts of remedies and tried everything from placing a hot iron on the heat marks over a cloth using steam, to not using steam, applying baking soda, baking powder, vinegar, oil, mayonnaise, I tried all sorts of remedies and nothing worked, in fact it actually got worse.
But then several months later I randomly googled "heat marks on veneer wood" and I came across a solution. The trick was to use a hair dryer directly on the heat marks. It did take some time and I had to keep at it for several days but it actually disappeared over time, even after applying direct heat.
This is it slowly fading away.
I used a Dyson hair dryer but you could use any hair dryer that puts off good heat. I had it on the hottest setting possible and sometimes applied direct contact to the wood but eventually it would over heat so you might want to keep it a little farther away from the surface. I would spend about 30 seconds per section and move slowly...
This is actually very easy that even a small child could fix my mess!
Final results! Isn't that amazing?
And just so you know, Hubbard's Maid Service does periodically offer "light house cleaning", which includes ironing but at this point you probably want one of our maids to do the ironing, not me!!!!