There have been threads discussing this in the past, and the consensus is that it's a bad idea. To carb a keg with dry ice, you'd be adding all of the CO2 to the beer at once. It would turn from solid to gas way faster than it would be absorbed into the beer, meaning the headspace would have to hold nearly the same amount of pressure as it would take to carb the keg in one blast. Hopefully the pressure relief valve would function properly and you'd just spray beer and CO2 all over the room instead of dealing with an exploding keg.
There's a guy here who's successfully carbed with dry ice in PET soda bottles. The trick is to cap the bottle and start shaking it extremely vigorously as soon as the dry ice is added. The shaking helps the beer absorb the gas so that the pressure doesn't get too high. You need to shake it very hard for several minutes or the pressure will build up until it explodes. I doubt the strongest person I know is capable of shaking a full 5 gal keg vigorously enough for 5 minutes straight.
There are much safer, easier, cheaper methods of carbonating, and I'm not sure I see any benefit of using dry ice. If you want fast carbonation with no risk of overcarbonation, buy a carb stone.