My house was built in several stage starting in the late 1800s. Throughout most of the house, the shiplap was nailed to the outside of the exterior walls and the inside was lath and plaster. In the stairwell to the basement, which runs along an outside wall, the shiplap is attached to the inside of the exterior wall and the outside only has the thinner wood siding.
I doubt that the purpose for the shiplap was primarily structural, but since there were no 2x4 braces in the corners of my walls and the walls are all still very plumb and square, I suspect it does provide some lateral stability.
You might want to consider blowing paper insulation into your exterior walls. It would only require you to remove the top-most shiplap board and reduce your risk of disturbing the lateral stability if you remove all the boards. Do you know what's under the exterior siding in your house?
One other thing to consider: If you do decide to remove all the shiplap, you may have to drill pilot holes when nailing then back up. It's not uncommon for older homes to have hardwood studs, and today's framing nails bend easily when hammered into old hardwood.