As Slomo said, if you run them dry you will end up with rubber Parts like fuel lines cracking earlier than you should and you will need a carburetor repair earlier from the needle and seat not shutting off and flooding out or migrating fuel over into your crankcase so the answer is no! Running them dry is not the best thing to do.
On a riding mower of the size you're mentioning and for a four to five month layout you really don't have to do much of anything because it's just not long enough and it holds enough fuel and passages in the carburetor are large enough that it won't make much difference but it's still best to fill them up with fresh fuel and if you're going to do that you might as well have that fuel be stabilized.
Now with the Honda mower it's much more important. The fuel shut off means nothing overall other than the fact you can shut it off and let it run for a minute or so until it actually dies out and you've got most of you out of there quickly.
Let me explain why this works better than someone just leaving a haphazard amount of gas in there or running them to what they think is dry from the fuel tank.
The problem with fuel is that is always in a process of evaporation and it always leaves crap behind nowadays.
So the problem is the more you have to evaporate the more it leaves. It's kind of like reducing something down on my stove and if you take raspberries or blueberries and do that you will see what starts out decently ends up being a gummy nasty mess by the time you're done. You certainly wouldn't want that inside your fuel system or carburetor yet I have found things that look just like that in the bowl of a carburetor like a mahogany colored sticky syrupy mess.
So when you run the tank dry until it stalls it still has more fuel in there and in the fuel line and then by the time you move it back around and it gets shaking All around they're just ends up being more fuel in the carburetor too evaporate down like this. When you shut off the fuel valve you're able to get rid of the vast majority of it right then and the small amount left probably won't leave enough of a concoction created problem and as soon as you turn the valve back on the deluge the fuel going in there will probably send everything out just fine.
But let's not forget that until the new overhead valve engines took over from the EPA law changing at the end of 2015 in the US, Hondas were the most finicky engine you can find.
On any smaller engine like that like on a push mower, generator. Pressure washer, tiller, log splitter, snowblower... At the bare minimum you should go by fresh gas the last time you're going to use the piece of equipment before you put it away and fill it full with fresh fuel and then when you're done top it off because that starts the clock from that day with fresh gas.
The much better thing and what I feel is the best thing to do is to add 1 oz of stabilizer for each gallon of fuel you're going to buy to an empty gas can a day as you go to the station to buy fresh gas.
Then get your gas so you're now have 1 oz of stabilizer to every gallon of fuel for instance if you're using sta-bil then go home and fill that equipment to the top of the tank and run it the whole time you need to use it and then when you're done top it off again and walk away from it.
With a Honda it doesn't matter to me whether you then turn the valve off or not but there is no reason to turn the valve off and let it run until it dies with this fresh stabilized fuel in there because the stuff is going to stay cleaner with that fresh gas keeping everything covered as opposed to the oxidation that starts to occur and the coating it starts to occur on the float when you drain the fuel off of those surfaces
