Trespass has many meanings. In the context of a national forest a trespass would be building a road, a house, digging a mine, commercial logging in a national forest without authorization or going into an area of a national forest specifically closed to public access for some purpose. Simply going onto national forest lands is not a trespass unless the area is closed for a specific purpose. The Forest Service has traditionally been extremely reluctant to close areas of the forests and when it does so, the closures are usually of very short duration. The most common reason is a forest fire is burning in an area. The last significant wide scale closures I can recall were during the 2002 fire season when the Forest Service closed all the forests in Arizona as all available firefighters were engaged in fighting fires and the Forest Service simply could not handle any additional fires. That year the BLM followed suit and closed the BLM public lands in Arizona as well. As soon as the rains came, the lands were re-opened.
Long term closures are used around facilities such as active mines and radio towers to prevent accidents and theft and around Forest Service housing to provide privacy to residents.