If oil or gasoline has gotten into the cylinder the liquid will not compress and you can't pull the starter rope.
Remove the spark plug and try to pull the starter rope. If it pulls easily, pull it 8-10 times and the piston will blow the liquid out of the cylinder. Place a cloth over the opening in the head where the spark plug was removed to catch any liquid that comes out. Once the liquid is removed, your snow thrower should start. This does not damage anything.
If you cannot pull the starter rope with the spark plug removed, it is possible the engine is worn and locked up. If the rope will not pull with the spark plug removed, try to turn the engine by hand. At this point, you might need to remove the outer cover of the engine to access the flywheel to see if you can turn it. If the engine is seized, you probably should have a service technician look at it. The service person can tell you for sure if the engine needs to be replaced.
If you require further assistance please resubmit question with details of the checks you have made. I will be looking forward to helping you again in the near future.