Not a good idea to shut down using keys with the traditional updraft carby, that will leave residual fuel in the carby venturi and induction area creating a back-fire risk on restart and fire hazard, as well as a kick risk if the prop is manipulated by hand while moving around. So no direct damage risk, but a few other risks that you need to be aware of. Obviously shutting down with mixture ICO will mean every last drop of fuel/air mix in the induction system will be sucked through the engine and most likely burnt as it dies as the spark is still happening.
The same risk of unintentional kick from manipulating the prop is there with fuel injected as well, as turning off the keys just stops ignition, so the fuel system is still charging. Have to remember vapor locks are caused by hot gaseous fuel in the system, not hot air, so residual fuel in the lines is the enemy, not air.
Not a bad rundown on what causes vapor locks and simple remedies, this comment was good;
Fred Sweet
Vapor lock also occurs in the fuel control unit and engines mechanical fuel pump. The mechanical fuel pump is easily heat soaked as it is mounted to the case. The electric fuel pump is typically remotely mounted and can still push fuel through the system. You can tell when fuel reaches the injector lines as the fuel flow indicator on many aircraft is located just prior to the fuel distribution block. If you prime the system with fuel mixture and throttle full open, the fuel flow will rise as liquid fuel flow past the fuel flow transducer. Shutting off electric fuel pump immediately after the rise in fuel flow will minimize flooding. At this point starting may still be difficult as the mechanical fuel pump is still heat soaked and can be vapor locked despite getting fuel into distribution lines. Beechcraft have a fuel return line so with the mixture in idle cut off, running the electric fuel pump for 60 seconds, fuel runs through mechanical fuel pump to control unit and back to tank cooling them off. Then a normal start procedure can be used. Many aircraft have a hot start procedure that is effective to the specific model of fuel injected aircraft. In the absence of a specific hot start procedure, if u have flooded engine, try cranking with mixture at idle cut off, throttle full open and gradually bring throttle back till engine fires, then gradually advance mixture. There are many hot start procedures, best is the one in POH if provided
The only addition I'd make to that is knowing which tank the return line flows to, if you have the other tank selected and the return tank is full you might end up pumping fuel overboard.