There is a basic misconception that I think you need to have clarified. An appliance does not simply “draw,” from the available volts, only the amount of volts that it needs. Rather, the available volts are going to impose themselves on whatever is connected to them, regardless of whether the load is capable of handling it. “Voltage” can be thought of as an amount of “push.” The higher the voltage, the stronger the push. What is being pushed is electrons, and the resulting current is essentially a measure of the flow of the electrons.
A common analogy used to explain the relationship between voltage and current is water flowing through a hose. Suppose you took a garden hose, one that is designed to handle water being “pushed” by the household water pressure. Suppose that instead of connecting it to an outdoor faucet, you bring in a high pressure pump, and connect the hose to that pump. What happens is that the higher pressure will be too much for the hose to handle, and the hose will probably rupture.
If you use a 240 volt source to “push electrons” through a component that is designed to handle no more than 208 volts, what will happen is similar to the hose rupturing. As Bob said, the appliance will overheat. If you are lucky, it will simply stop working. If you are unlucky, it will set the place on fire.
Therefore, it is important to know for certain whether the appliance has been rated, by its manufacturer, for both voltage levels. Also, it is important to be certain of the voltage level that your building actually uses. As sparky mentioned, 240/3 phase is rare.
Charles E. Beck, P.E., Seattle
Comments based on 2014 NEC unless otherwise noted.