In principle, the data stored on a hard disk will affect its weight. The issue here is energy: since energy and mass are in some respects equivalent, both experience gravitational forces and both have weight. The more energy a hard disk has stored in its surfaces, the more it will weigh.
It's not clear to me whether a full or empty hard disk will be heavier, because it depends on how the surfaces are magnetized and on how data is represented by those magnetizations. If we assume that the disk is designed so that the tiny permanent magnets on its surface are magnetized in or out of the plane of the disk, then the highest energy magnetization will occur when all those tiny magnets are aligned with one another. They'll then have lots of repulsive interactions and few attractive ones and the total potential energy stored in the disk will be large. The disk will then weigh slightly more than at other times.
But don't worry about someone "reading" your hard disk by weighing your computer. The weight changes that we're talking about here are so incredibly small that it's unlikely they'll ever be detected, let alone studied in any detail. Nonetheless, it's an interesting question and there are situations in which stored energy is large enough to weigh.
Answered by Louis A. Bloomfield of the University of Virginia