The reason to store them with dry carburetors - both 2 and 4 stroke, but especially 2 strokes - is because if you don't, crud forms in the carburetor during storage. In the case of gravity feed 2 strokes, there is a good chance fuel will evaporate from the main jet area, leaving the oil behind, so the jet is more or less blocked by oil when you next try to start them. Of course if there is ethanol in your fuel, it will absorb water, turn yellow, and be unsuitable for further use. On top of those factors with diaphragm carburetors, even good fuel (no ethanol or ether) gradually attacks the diaphragms which brings forward the time when you'll have to clean them out and put kits in them.
In the case of suction carburetor engines such as the small Briggs ones, there is no fuel tap, no continuous feed of fuel into the carburetor when the engine is not running, and no oil in the fuel. The only adverse effects of not draining the tank and running them dry is that gum may form in the carburetor, plus in the case of Pulsa-Jets or Pulsa-Primes the fuel pump diaphragm will continue to age when you aren't using them. With Australian fuel (no ether), and as long as you avoid ethanol-containing fuels, in my view it is a lot easier to replace the diaphragm every ten to twenty years than to tip the mower upside down to drain the fuel after each time I mow, so I just switch them off, trundle them into the shed, and don't give them another thought until next time I mow. With Honda mowers I turn off the fuel and shut them down with full carburetors because the float bowl holds so much fuel, and they use it so slowly, that it otherwise takes forever to shut them down, and it doesn't seem worthwhile to run them dry. Bear in mind that those Honda carburetors do not have diaphragms, all that happens if you leave fuel in them is that you have to remove the main jet and emulsifier every few years to clean the gum out.
One thing I do not do, is run 4 stroke carburetors dry with the engine under load. Lean mixture with the throttle open causes high combustion chamber temperatures, which means high exhaust valve temperatures. I'm also careful to keep the throttle nearly closed when running a chainsaw carburetor dry - chainsaw engines lead a pretty marginal existence at the best of times, without adding lean mixture at high speed to their troubles.
