I don't know of any Forest Management Plan that explicitly prohibits trail-cams from being place in a designated wilderness area. There are other rules that apply, where this could be construed. This is founded in the Wilderness Act, where the intent was to preserve lands in a pristine condition, with little to no sign of humans. In that, there are the "leave no trace" concepts that apply. Leaving a trail cam behind, and geocaches as another matter, would apply under this principle. But trail cams were not around when the Wilderness Act was passed, and are not prohibited by that law. The prohibition would come from the Forest Management Plan for that particular National Forest. And again, those usually don't get that specific.
As far as needing a permit to use one on public lands, that would apply if you used the pictures for commercial purposes. Any activity conducted on public lands, and that includes National Forests, National Parks, BLM Lands, Wildlife Refuges, National Monuments, etc.... that is commercial in nature, is required to get a permit from the appropriate agency. Activities that would need permits include mining, timber, grazing, rock collecting, outfitting/guiding, photography, bird watching tours, etc.... If you make money from it, you have to have a permit. For something like photography, the permit is a simple form you put your name on pay your $10 for your "special use permit".
I hope this helps.