Assuming you are a somewhat healthy individual and not camping in an avalanche zone, you probably do not need to worry about suffocation. With enough accumulation, your tent will collapse, at which point you will wake up and most likely think you are being abducted by aliens; it will be completely dark, there will be an unnatural amount of pressure on your body (from the snow), you will not be able to move (you will be trapped in your mummy bag), and there will be a cold wet piece of material on your face (the tent). I am pretty sure there would have to be something seriously wrong with you to be able to sleep through having a cold wet tent on your face; it really is unpleasant. If the collapsed tent persists long enough, eventually there will be a C02 build up which will cause you to wake up gasping for air. Note that CO2 is not CO and is not deadly in the same way.
After you wake up, you will likely scream and panic for a few moments (remember you think you are being abducted by aliens). Then you will realize what is happening and you will begin to extract yourself from your sleeping bag and tent. This generally isn't all that hard since the tent often keeps some of its structure. Once you are out of your tent, it will be a race to find your gloves and boots (and possibly pants) before hypothermia/frostbite sets in.
If your tent sheds snow really well, it is possible to end up in a deep hole with snow going up the walls, but again suffocation is not really a concern. Eventually the CO2 levels will rise and your body will do the thing it normally does during high CO2 levels (gasp for air). At this point you need to dig yourself out and clear off the tent.
The best thing to do during heavy snow fall is to wake up every couple of hours and shovel off the tent and around from the entrance. You can keep the snow on the sides as it will provide warmth like a snow cave.