When I worked for the Forest Service during the last three government shutdowns we stayed open. For the first two I even had to show to work in spite of being a non-essential employee!!
However, it appears that this administration is trying to make a point about the shutdown and making the Federal agencies toe the line. All Federal agencies with a couple of exceptions work for the President. Here in Washington state, the Park Service, is closing boat ramps in the middle of nowhere. The Park Service has always over-reacted to political events.
The closer you are to urban areas, the more likely it is that the Forest Service and BLM will follow closures since those sites are so politically visible. That is, the politicians notice and put pressure on the agencies to close public lands.
Out in Utah, and more rural areas the pressure is on the other side of the fence. So those areas will tend to stay open. To "legally" close public lands there have to be closure orders written and signed and POSTED. It is difficult to do in the event of an emergency like a fire.....but I guess political emergency is MUCH MORE IMPORTANT.
The one exception is the National Park Service. They are much more of a centralized and political follow the rules from their Washington office sooner rather than later.
The National Forests around my house are still open....but the office is closed.
Here is an article from Montana on the closures: http://missoulian.com/news/local/u-s-forest-service-workers-head-home-national-forests-remain/article_7abbed24-2ab7-11e3-bba4-001a4bcf887a.html