I know you are not in Arizona (I am) but here is the AZ BLM answer on the subject.
Thank you for your recent questions regarding recreational vehicles (RV) and dispersed camping on BLM lands in Arizona. The answers to your questions are more complicated than originally thought. Although the Code of Federal Regulations (CFR) found at 8365.1-1 (3) generally excludes 'wash water' from BLM's prohibition against draining or dumping, it can be specifically prohibited by Supplemental Rules issued for a specific area. This applies equally to RVers and tent campers.
TITLE 43IC LANDS: INTERIOR
CHAPTER IIAU OF LAND MANAGEMENT, DEPARTMENT OF THE INTERIOR
PART 8360_VISITOR SERVICES
Subpart 8365_Rules of Conduct
Sec. 8365.1-1 Sanitation.
(3) Drain sewage or petroleum products or dump refuse or waste other than wash water from any trailer or other vehicle except in places or receptacles provided for that purpose;
There are two locations in Arizona where draining wash water is specifically prohibited by Supplemental Rules that have been established and were published in the Federal Register. They are the Long Term Visitor Areas outside of Yuma, Arizona, and Hot Well Dunes Recreation Area east of Safford, Arizona.
A note of caution to your audience: Under State laws and regulations in Arizona, 'wash water' or 'gray water' from a kitchen sink or dishwasher is classified as sewage. If discharging it onto the ground from a RV or camper might cause it to enter an aquifer, the visitor could be subject to violation of State of Arizona regulations unrelated to BLM regulations. Even if the gray water is from a clothes washer, bathroom sink, shower, or bathtub, it can only be discharged if done so according the 'General Permit' practices that would apply. The practices are explained at the Arizona Department of Environmental Quality.
In addition, if the gray water creates a hazard or a nuisance a Law Enforcement Officer can cite (or in extreme circumstances arrest) an individual. This would go beyond simple gray water dumping, and the citation would likely be for some other offense related to degradation of resources or public health and safety issues. Law Enforcement Officers in the field have discretion in applying the laws and regulations as necessary and appropriate to protect the natural resources on the ground.
Thank you for your patience, while BLM researched the issue in order to provide accurate answers for your audience,
Carrie Templin
Public Affairs Specialist
Bureau of Land Management
Arizona State Office
(602) 417-9448