Insulate panel-style board and batten walls with extruded polystyrene (XPS) rigid foam, usually 1 1/2 inches thick. Use unfaced XPS, with fiberglass batts that have a vapor facing to be installed inside the wall. Nail 4-by-8-foot panels to wall studs with a hammer and 3-inch galvanized nails, to penetrate 1 1/2 inches into the studs. Put siding panels over the XPS, and then nail to studs through the XPS, ensuring you don't dent or compress the XPS.
Use polyisocyanurate insulation for board and batten siding fastened to horizontal furring strips, nailed across studs, to provide nailing points for boards too narrow to be fastened to studs. Fit polyiso sheets horizontally between furring strips. Fasten polyiso to furring strips through tabs on the side. Avoid nailing into or tearing the polyiso covering, which must remain intact to preserve its heat resistance.
Install unfaced fiberglass insulation on the inside of polyiso walls, to prevent having competing vapor barriers that would trap moisture inside the walls. Allow about 1/8-inch gap between boards or panels to permit some expansion. Nail vertical battens, typically 1-by-2 or 1-by-3-inch boards, over the seams between boards or panels, using nails long enough to penetrate into the studs or furring strips. Avoid nailing into polyiso between furring strips, to keep it intact.