It depends on the type of hard drive. Magnetic drives for example can’t change mass like that, all the 1s and 0s are are just an orientation of atoms.
In something electronic like RAM, the data is held using a feedback loop, electrons flowing through wires and transistors in such a way that they keep flowing through the same channels unless the channel is diverted externally (changing a 1 to a 0 for example). In such a circuit, there’s a higher concentration of electrons flowing through the memory cells storing a 1, in that sense, those memory cells weigh slightly more.
So, in a very real sense, 1’s are heavier than 0’s.
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