Oh man I have first hand experience on this topic: Im in Alaska. It costs $100 to get a filming permit on USFWS lands and in some cases for BLM the price starts around $500 due to the EIS process. If you plan to make a profit from the film, even monetized Youtube accounts, a permit is required. That means the production video and its content if used commercially requires the permit and up to a 6-month wait to get it for commercially filming on federal public land. Each agency has different requirements, timelines and fees.
However, if you don't charge for viewing your video, you don't need permission or a permit. Additionally, you should remember that the BLM agency automatically notifies any neighboring land owners about your intent prior to you going there... I was filming Project Bloodtrail in 2007, which was a covert hunting study navigating a public lands river surrounded by native lands. BLM didn't tell me upfront they were obliged to notify the land owner, and that native corp sent scouts to spy on my operation and caused some conflict in the field about land boundaries and our presence "on their lands." They even requested trespass enforcement by the State Troopers which wasn't enforceable on private lands. None of that sat well with me, even if on the right side of the law.
In 2006 I was cited for 2 counts of filming without a permit on federal lands for the same video. Each fine was $1000, and one of the clips I got cited for was a 30-sec B-roll clip from my buddy's hunt on a different river on different National Wildlife Refuge. Hide officially chapped!
I have story after story for examples of federal management staff surfing the web and discovering "suspiciously familiar landscapes or identifiable terrain features to nab a scheme or create a "case against" for disturbing in situ artifacts or horn/antler length questions, you name it. I stopped getting permits and just don't charge for content, mainly so I don't have to disclose my plans to the land managers what I'm doing and where on public lands.