Milt,
I agree it will rotate but not translate i.e., move in a linear sense.
Assume the length of the plank is 2d, the applied moment is F*d whereupon you will get an angular acceleration.
Now if I apply an opposing moment to prevent any rotation, of magnitude F*d (Note: this doesn't affect the force equilibrium), I now have neither a rotation or a translation.
Agreed?
Isn't this case directly analogous to the one in question?