Every gas furnace needs a steady supply of air in order to properly work -- otherwise the combustion chamber will not have enough oxygen to create a strong flame. If the vent becomes covered with snow, air will dwindle until the furnace will not be able to function correctly. It may start and then quickly shut off or continue work but fail to heat the house properly.
While problems can occur when snow blocks the furnace vent, removing the snow can create problems as well. When the vent is covered by a snow drift, it can be very difficult to tell where it is. Shoveling away snow with a shovel may only slightly damage a vent, but using a snowblower can tear the vent apart. Either way, you may need to pay for repairs or service to the vent.
When furnace vents are covered, even vents on a rooftop, the exhaust that the furnace creates has nowhere to go. This exhaust can eventually leak back into the air ducts and start spreading throughout the house. Not only is the exhaust filled with contaminants from the burnt gas, it also has trace amounts of carbon monoxide and can be unhealthy or dangerous for anyone living in your house.
Snow also creates a furnace vent problem because it is continuous. While you can clear the snow away to prevent any major problems, if you live in a climate in which snow falls throughout the winter, you will have constant work to do. The primary solution is to restructure the vent to get it away from the snow or to add a hood to protect the vent opening. Both options can work but are an expense.