Not sure if you mean smoke from tires that are made from clay or the smoke would be made from clay.
I would think the tire material would be unimportant. For the smoke, rather than clay I would consider cotton. Another thing occasionally used is angel hair used for Christmas decorations. Getting hard to find these days but it is more translucent than cotton. Cotton is pretty much the standard way to make white smoke.
I have a similar need, and will probably use cotton dyed with a tan dye. I want to make a diorama with a couple of midgets on a dirt track, throwing the dirt out behind the rear tires. The dirt spray looks a bit more granual than smoke, however, so if anyone has ideas other than tan-dyed cotton I'd like to hear it.