I went looking for second opinions. The magic of second opinions is that if you ask for enough of them you will actually get the one you wanted to hear. I found a knowledgable sort who said that it would be OK to wood-burn on MDF, as long as I did it outside or in a well-ventilated area. However (there is always a however) the MDF might melt before it burned, and even if it did burn, it probably wouldn't burn evenly. I set out to test this. First I needed a piece of scrap MDF. This turned out to be trickier than it sounds. The saws at both Home Depot and Lowe's were broken and it is hard to take home a six-foot piece of lumber in a Miata. Eventually, the folks at Lowe's found a broken piece for me.
Here is a picture of my test burning. Not as nice as burning on wood, but it will work. After all, the wheel will be turning most of the time, right? Pretty stinky, though. After that I launched into burning other stuff.
Here's all the stuff I have done so far. The box-like thing is the cover to my doorbell. As you can see, I am going for a primitive, folk art look. My inspiration is the bright-colored Mexican folk art I remember from growing up in Tucson. The motif for the wheel will be vines and snakes. I am still working on a source for the color.