Nuclear Energy Reactor Demonstration Act This bill directs the Department of Energy (DOE) to advance the research and development of domestic advanced, affordable, and clean nuclear energy. Specifically, DOE must enter into agreements to (1) complete at least two advanced nuclear reactor demonstration projects by December 31, 2025, and (2) demonstrate at least two and not more than five additional operational advanced reactor designs by December 31, 2035. DOE must establish goals for research relating to advanced nuclear reactors, including by coordinating with members of private industry to advance the demonstration of various designs of advanced nuclear reactors. DOE must also carry out a program to conduct research relating to (1) next-generation light water reactor fuels that demonstrate improved performance and accident tolerance, and (2) innovative advanced reactor fuels that demonstrate improved proliferation resistance and use of resources.