All the information I could find, seems pretty vague on what impact the government shut down will actually have on Dumont. Dumont is not a National Park, so as far as I can tell, there will just be no BLM officers, no BLM services, the bathrooms locked and maybe some signs posted saying closed. I can't find any information on if there will be fines if you are caught on "closed" lands, however, with no services who is going to catch you? But then further down it says they will be on duty. And yes of course the BLM sites are down, making it even more difficult to find any information. Its not like they are going to run out there an put up a fence and a gate across the road and lock it up, that would require BLM employees to work and funding they apparently don't have.
http://www.doi.gov/shutdown/fy2014/upload/BLM-Contingency-Plan.pdf