Dilute (5% or so) HCl should work, though it might corrode nickel-plated tub fittings. Better would be to use commercial lime/scale remover, agitated with a plastic utensil in a plastic tub placed inside the bathtub, being careful not to get the solution on clothing, floor (acid dissolves grout), skin or in eyes!
However, consider the economics of this. A shower-curtain liner locally costs US$3 to US$6. A bottle of lime remover (dilute HCl) costs US$1.30. How much additional use will you get from the liner after cleaning? Liners often tear while being installed or removed from hangers.
Perhaps a more useful plan is to repurpose the plastic liners. They can be put under crushed stone to prevent weed growth, they can be placed on a vehicle(or even a driveway) to ease snow removal, etc.