It will change the function somewhat, depending on the thickness of cob, in that there will be less immediate heat radiating from the barrel and more stored heat radiating later.
It will also reduce the difference between the heat riser temperature and the barrel volume temperature, lessening the draft production by some amount. I have heard reports that this can cause a problem, but if you otherwise have a chimney with good draft, you will probably be fine. It all depends on the details of the particular system. Some apparently work very well with a cob covering.
Those are the largest trousers in the world! Especially when next to this ad: