You will be doomed!!! Just kidding. Probably not, but you will significantly increase your chance of ended up in a ditch. I drove a high performance AWD sports car through 3 winters with moderate to heavy snow on summer tires (never ended up in a ditch or had damaged body parts BTW, just swapped on Blizzaks for this winter), and I can tell you that while it is doable, there are a few things that you need to be extra cautious about.
Snow tires are good, and they are much better than summer tires or all seasons in terms of grip on snow. But they are no magical tires and cannot defy the slippery property of snow, so that means while you have a bit more grip when comparing with the summer or all season, you still have much less grip on snow when comparing with snow tires on dry pavements. So you still cannot drive like you are on dry pavement or in rain when you on snow, and following suggestion still hold.
Check the grip threshold every time the road condition changes (change in snow coverage, type of snow). Here is how: brake and increase brake force smoothly and slowly until to the point of when ABS start to kick in or tail starts to slide sideways, and back off smoothly. That tells you how much brake you can apply without losing control. My car's tail start to slide sideways on snow if I apply the brake for more than the initial 1/8 of the total pedal travel. Check your mirrors and don't do that when someone is right behind you, unless you seriously want that person to ended up in a ditch :P ABS is a good thing when you freak out and cannot let go of the brake pedal because it forcefully releases the brake pedal in quick succession so you can have extra grip for steering. But it is not as effective as easing off the brake to release grip for steering yourself.
Keep driving on asphalt whenever is possible (drive in the "black trench" dug out by the car in front of you), also when you are in a corner, keep the outside tires on the cleanest surface possible if the clean surface is not wide enough for you to keep both sides of the tires on it at the same time. More weight is distributed to the outside tires when you are in a corner. That means you need to know more or less where exactly your tires are on the road when driving.
No erratic movements at all. Even during emergency evasive maneuvers, and limit your steering and brake travel to as little as possible because the amount of grip is limited, and the amount of grip used by steering cannot be used by braking and vise versa.
Keep a monstrous amount of distance between you and the vehicle right in front of you. I mean a MONSTROUS amount! I'm not talking about 100 yards or 200 yards. I'm talking about 500 yards or more when on highways.
With summer tires, avoid slopes (err did somebody say ski hills...?). Especially stopping on slopes. All season still needs to be aware of this. Chances are once you stop on a uphill, it is really not very likely to get going uphill again. And downhills, you should probably not think about trying to stop on a fully snow covered downhill unless you have winter tires.
You can pretty much forget about passing, switching lanes and driving at the speed limit. Normally there is a constant stream of snow pile in between lines on highway. It is a good way to spin out and visit the ditch if you are not smooth enough when attempting to cross that.
If in any case your car start to lose control, DON'T PANIC! Back off doing whatever you were doing smoothly to regain grip (if you are steering, turn steering wheel less; if you are braking, brake less). Panicking will cause you to steer fanatically and brake heavily, which cause you to lose control over the car more, and possibly into a ditch.
Edited by LaserPower - 11/3/13 at 4:07pm