I think that due to the fact the heat wouldn't be distributed evenly that a small hotspot would smoulder in the bottom of the cup which the water would then leak through. If anyone does this as an experiment could they let us know what happens!!
When I was a boy, my dad took a sheet of paper and held it so that it could hold a small puddle of water, he then carefully lifted it over a candle and lowered it onto the flame until the water boiled, the paper did not burn.
i think you should just get a paper cup, fill it with water, put it on the barbecue, watch, and then tell us all what happens! if not, i will try next time i am barbecueing!