Are there dangers to using solid fuel on a propane grill?
Propane grills mix air with gas before burning it.  Ash from charcoal and other contaminants can clog the flue and produce inefficient burning of the gas, possibly produce carbon monoxide (poison) or even gas build up.  

As long as the gas parts (igniter, flue, pipes, etc) are clean and unaffected, the charcoal, hickory stick, etc, should not cause any issues to my knowledge.  

On the whole, this idea is more on the terrible side than great.  A charcoal BBQ is relatively inexpensive and with the aid of a hair-dryer can light up in little time :)