What trade handles old wood paneling?
I would typically assign something like this to an interior trim carpenter, but it depends a lot on the quality of their work.  In my experience  I would avoid the painting company that also does trim.  They are fine for baseboard and crown, but for this job I would want the interior trim carpenter that does built in entertainment center cabinets, wainscoting, and things like that.

Again, It depends a lot on the quality of their work.