The FAA furlough of 3,000 aviation safety inspectors and the NTSB furlough of accident investigators may the United States in default of its treaty obligations under the Chicago Convention and the obligations of member, states of the International Civil Aviation Organization.
According to a Forbes article on the subject, the Chicago Convention is the main aviation treaty that sets the standards for air safety among the 192 contracting states of ICAO, and violating the obligations spelled out within it, could have significant repercussions for US airlines if the furloughed employees are not put back to work.
ICAO requires the U.S. to have a system for insuring that all aircraft operating over its territory and all U.S.-registered aircraft comply with safety regulations, one of which is having an adequate number of aviation safety inspectors.
The U.S. is also required by ICAO to ensure that persons who violate air safety regulations are prosecuted and to conduct accident and incident investigations. Furloughing NTSB accident investigators could violate this requirement.