Some places may require a camping permit to camp in a national forest while other places ma not.
I've camped in a few national forests with my small travel trailer and stayed for a couple weeks and no one bothered me about it.
When camping in a national forest you must move at least 5 miles after 16 days to another camp area in the national forest.
I do know though that if a group of over 75 people wishes to use a national forest to camp then they must obtain a special use permit which is free to get.