It isn't that the motor won't run, but that a) the efficiency and life of the motor will likely be reduced because b) the current draw will go up at the lower voltage.
The speed of induction and synchronous motors (which most 240V tool motors will be, single or three phase) is determined primarily by the line frequency. The power needed by the motor is determined by the load-the motor doesn't care how much current it draws... it draws what it need to to meet the power demand of the load at the run speed. At the lower voltage (about 12%) the current will be higher (again, about
12%), leading to greater heat production in the motor and greater I^2R losses in the motor and supply wiring. If the load is near the 240V rating of the motor(compressor motor, large power tool, etc) then at 208V, the motor will like lose some or all of it's magic smoke and cease to function, especially if that 208V supply is really only 200V (5% either way is very common with system load variation, 10% not unusual, especially in the summer when lots of AC units are on).Motors are generally not conservatively rated-it isn't economical to overrate. You get the 10% or so maximum supply variation built in, and that's it. Go below that, and you need to begin derating the motor rapidly, go above that, and the likelyhood of insulation failure goes way up.