Anyone with a tool for removing staples can do this. If the fabric is old cotton cambric, it's probably on the fragile side. Black dye weakens cotton fibers over time. Unwoven synthetic "cambric" has been used for quite awhile now (as in decades) and that stuff can tear too but it's easily replaced. Some upholsterers use woven cotton fabric instead of the non-woven synthetic stuff---also easily replaceable. I don't know if actual cambric fabric is even made anymore. (Cambric is the type of weave---when used on furniture, it's called a dustcover because it keeps the dust generated by springs rubbing against burlap and padding from sifting onto the floor.)
Anyway! Go for it. If you only have an electric or manual stapler, just don't re-staple really closely as the staples are heavier gauge and may split the wood if they're close together. And what Leif said may be true though I've seen dustcovers on furniture with screws, too. I think it depends more on springs, webbing, padding, etc.
If you want to replace it with woven fabric, just go to any fabric store and buy lightweight black cotton which will be with the quilting fabrics, or get synthetic cambric where upholstery fabrics are sold. Or buy either one online.
If you need any help, please contact us at – info@designaddict.com