The shutdown should only affect NF land where you need a permit or have gates on access roads. For example, you do not need permits in the Wind Rivers (Bridger and Shoshone National Forests) and many access roads to trailheads are county roads.

I personally would still do a trip as long as I could park my car on non-federal land. For example, Twin Lakes trailhead to Hoover Wilderness is on private land. The private owner does charge to park, but I would still go in for a backpack, even without a permit. With such limited resources, giving you a ticket for going in without a permit is the last thing on the mind of the Forest Service.