My snowblower starts up easy but then it quits

If it's the starter that makes a humming noise then you need to check the starter out. the gears could be stripped or even a tooth missing so it won't grab the fly wheel. Try pulling the start rope and see if that makes a difference. My Craftsman 8 horse has a recoil starter as well and I quite often use it as well. I don't always use the electric start.

Stuck? you mean if you try to use the pull starter the engine will not let you pull the cord out?
Will it not start OR is the engine frozen. help us with your description
Take the Spark Plug out and see if that helps it turn over. Maybe the float in the Carburetor is stuck open and the cylinder filled with gas Causing Hydrolock.

Possibly the friction wheel or drive belt need to be replaced

Try using a spray starter like that used to start a car engine with a carburetor.

check out the interlock switch system, check battery with a load meter first to make sure battery is up to snuff

Any engine that burns oil is not a good sign. Often, it indicates and engine that has either been abused (overloaded repeatedly a.k.a. "beat"), is aging or wearing out. Sometimes, overfilling the crankcase can cause this too. You may put off engine failure by using a thicker oil - but that can make starting difficult in cold weather as the oil will be *really* thick or you can try using an additive designed to help control burning oil. Your local auto parts store will likely have quite a few products from which to choose.

Good luck!

Most of these engines have a low oil switch to keep them from burning up. Fill it to proper spec and try again.

Clean the carburetor and check the intake manifold to the head for air leaks/bad gaskets.