Use some starting fluid and see if it starts and dies. If it does
you have a lack of fuel.
Turn the fuel line off (closed position)
Take the carb bowl off and clean it all out with gumout. (or something
similar) Clean around the whole carb with it actually.
BTW if the carb is all gummed up and look similar to tar, then you
got more serious issues. If that's the case I found it simpler to
remove the carb and either A) Get another carb off of ebay if cheap
B) get a carb rebuild kit and go at it C) Get all new gaskets only
a carb boil kit at your local auto store.
When going for the long term storage solution... It all depends on
how far you want to go. What I do is run the engine (with the gas
tank empty) until it sputters. Put the choke on and it will run some
more. After it dies, some carb bowls have a drain on it. Empty out
all the gas if any left. (with the carb bowl off or open, make sure
fuel line is open and all the gas is out of there.)
If the gas tank on it is metal, then I use a light spray oil (wd40 or
something similar) to coal the inside of the gas tank.(prevents rust,
not needed if you have plastic)
You can put a little fogging oil inside the cyclinder (take out plug,
clean it up or replace) and turn the unit over a few times with the
ignition set to off. This will coat the walls of the cylinder and
the piston itself.
Drain and replace the oil if it has been in there for awhile.
(used oil can be acidic and cause damage just sitting there)
Put it all back together, maybe even spray down the
carb and its linkage (springs etc etc) with some oil and
that should be it. With any luck there shouldnt be little if
any rust and should start up next year with minimal effort.