I wouldn’t. You can create moisture and mold problems, much less drastically change the size and character of the interior of your home.
Adding an insulated wall on the interior of your home will change the “perm” rating of your exterior wall. Perm rating is important because it controls the flow of VAPOR through the wall.
Depending on where you live, vapor will travel from the inside out (northern climates) or from the outside in (southern climates) or both ways (middle climates). When vapor travels through the wall, it changes from a vapor to a liquid. If this liquid gets trapped in the wall, it will create rot and mold. Here is an article that explains it better.
All materials have a perm rating... even a coat of paint. Here’s an article that lists some materials and explains it better. Adding additional materials could trap moisture within the two walls and create rot and mold.
If you’re trying to save heat and reduce your heating costs, I’d concentrate on 1) adding insulation in your attic, 2) sealing drafts around windows and doors, etc., and then 3) insulate your exterior walls.
Heat rises, so concentrating on the ceiling makes sense. (The Department of Energy has a website that explains this.)
Drafts will lower the room temperature dramatically and keep your furnace working.
Yes, due to the age of your home, I doubt if they have any insulation in your exterior walls. Adding insulation and getting thermal windows (or storm windows) will help...but not as much as 1) and 2). (Besides, a new 2x4 wall is not that thick so you won’t be able to add much insulation.)