How well does a chemical peel work for removing old scars on the face?

Depends. Chemical peels and/or laser treatments may be an important part of scar revision but without knowing more details, detail recommendations are impossible. The size, location of the scar as well as it's character (flat, heaped up, etc) all influence the best way to improve the appearance of a scar. Other treatment options include dermabrasion, scar revision surgery, and injections.

Related Questions

Will a chemical peel help with the pitted scars on my face from acne?

In some degree. Chemical peel can reduce the appearance of pits to some degree. For deeper pits a minor surgical intervention can be needed to with injection of fat or fillers under the pits.
Partially. Use of chemical peels follow the existing surface scars so cannot provide much smoothing effects. Mechanical dermabrasion to plane the skin usually works better. Some strong laser resurfacing works. Each help stimulate new collagen under your skin to provide some additional smoothing of the scar sites.
A little. Deep acne scars are difficult to completely remove. Chemical peeling will help some, but the best results are with fractional laser skin resurfacing. It usually takes 2-4 treatments depending on how bad the acne scars are, but the improvement gained with laser treatments can be impressive.