The USFS knows that some visitors to Deep Creek (probably a majority), go through Mike 's gate to park and access the springs. They have told Mike of the closure. Rick and Wizard are right, Mike could easily be sued by an uninformed Deep Creek visitor who passes through Mike's gate if Mike does not tell the visitor about the closure. Let's say that person does make it to the springs and gets cited by USFS. That person has every legal right to make Mike pay for the citation. (Collecting from Mike is a different story ;-)
Mike would not be responsible if he told a visitor at his gate about the closure and the visitor drives down JP4 and/or accesses the springs via another route.