I dont' quite understand, you don't want it to use ink at all correct, just burn the paper essentially?
If this is the case it shouldn't be too hard to make something. Think of a flashlight and that sort of switch setup where you can open and close a circuit. Now the only thing is that you want heat generated, so create a circuit using thin copper wire...when it closes it should heat up if I'm not mistaken, then you could "write" with it and open the circut again when you're done.
If this is not what you mean, excuse this idea.