The National Register of Historic Places is part of the National Park Service (NPS). From a federal perspective, a property owner of a property listed in the National Register can do what they want with their property if no federal funds are attached to the property. (For more information, visit NPS online.)
If federal funds are attached to the property, however, then any changes to the property have to allow the Advisory Council on Historic Preservation to comment on the project.
State historic preservation offices (SHPO) oversee historic preservation efforts in each state. There may be state or local preservation laws that the owner should be aware of before undertaking a modification, remodeling or renovation project with a historic property.