Nope, the working drive will not break in the non-working computer as far as i know, i dont see any possible thing that could make it do this.
But even by chance that you kill it, which is highly unlikly, they are VERY cheap to buy. So dont sweat it, go mad, take it out and put her in the broken computer, and follow the steps i listed above, should work out fine for you. if not post here and we will see what to do from there