I do not believe that being on National forest land will exempt you from the county leash laws. t he forest is inside the county line then it is under the sheriff's jurisdiction

one possibility of a fine you could incur:

Permitting a dog to run at large is a violation of Park County Resolution 95-11-8

Permitting a dog to chase wildlife or livestock is a violation of the law and carries a penalty of up to $500 with a mandatory 10-day quarantine.