My experience has been that you almost always need a permit to enter wilderness managed by the National Forest Service or the National Park System, even in winter.
These regulations are done on a case-by-case basis. For example, in Colorado, the vast majority of wilderness areas on Forest Service land require no permits for entry or dispersed camping. In fact, no permits exist. The exceptions are the relatively small number of wilderness areas that experience particularly high visitor volumes, which are the ones that I avoid anyway. Most of those just involve filling out a form at the trailhead, and a few extremely crowded areas require reservations (Indian Peaks, Maroon Bells, etc).