The best way to save money on a demo project is to do some of the work yourself. Simple projects like removing wallpaper, pulling up carpet, or taking down closet shelving are easy DIY tasks and not worth paying a professional to do. Instead, save the heavier or more labor intensive jobs for the paid contractor. Projects like removing countertops and cabinets requires several crew members and can cause damage if not done properly.
Another way to save money on your project is to salvage and resell your old cabinets, countertops, or lighting fixtures that are still in good shape. If you don't want to resell them, consider donating the materials and appliances to a charity like Habitat for Humanity where they will be of use to a family in need.