You can try it, the idea of any air cooling whether air cooler, radiator cooled by air etc is just to exchange the air. Move coolest possible air (ambient) over the hot components to cool them. If the air in your gaming room is hot then that's the only air you have to work with. The air inside the case will likely be around the same as the air outside the case so long as the air is moving through the case freely on a regular basis. It certainly can't be any cooler than the ambient air in your room. Since a case isn't totally sealed, air is going to exchange between outside and inside your case one way or another. Whether you're using the fan to actively pull room air into the case or trying to push case air out (room air will then find it's way into the case as a result).
I think you'll be better off leaving the fans as intake. The components inside the case are what need cooled. Drawing air out with the fans exhausting, all you're doing is creating a single point of air exiting the case, at the fan(s). As intakes, they can blow air around, it will hit various components and bounce off circulating the air in the case to help eliminate air 'dead' spots.
You can always turn them around and try it, monitor your various temps (cpu, gpu, chipset etc) using software like realtemp and hwinfo64 or similar. Make a note of what your average temps are with the fans as intakes, then mount them as exhaust and see if the temps change any. Your best bet to improve cooling is cooling down the ambient air in your room since that's the basis for cooling all the components. Either an air conditioner, fan, something.
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