I painted my stairs and then applied a coat of gel stain. A week later, the surface is still not dry. How long will this take and is there a method for hurrying the drying process?

You put a Gell Stain OVER paint? Why? Gel Stain is just a non-runny stain used to color bare wood to a desired tone... its not meant to go over paint.. or be a finsh coat on anything.

Gelcoats are epoxy or resin based.

I doubt that it will ever dry. Wait another week and then strip it with 5F5 and scrapers and steel wool.

When I was little in the 50s I painted a picture on a glass jar with exterior house paint and then applied a coat of shellac (resin based) before it was 100% dry, and it never ever dried.

(I do believe that gel stains are suggested by paint stores for going over OLD chipped paint like often found on stairs, because it's supposed to cover all that with the least amount of work. Not over new paint!)