What is the danger in running a window-mounted air conditioner on a carpet floor in an enclosed room?

So I take it you mean setting the AC on a table or chair or flat on the floor? The only trouble is the condensation the AC creates as it cools the air. Depending on the humidity of the air this could be a little water or a lot. I would suggest to at least put the Ac up on a table and run a hose from the rear water tube to a sink or toilet or something to collect the water.