What permits are needed to operate a human-flying drone in the United States?
According to https://www.faa.gov/uas/beyond_the_basics/#55:


  To fly a UAS that weighs 55 lbs. or more, operators will need to use the existing Section 333 exemption process.


The Section 333 page has further links.  Spot checking some of the approved exemptions shows that the exempted status tends to look quite a lot like the 14 CFR 107 (sUAS) rules and include phrases like "carrying no passengers or crew", "weighing less than 55 pounds including payload".

As far as pilot certification would be concerned, exemption 11422 (which is one I spot-checked with) says:


  Under this grant of exemption, a PIC must hold either an airline transport, 
  commercial, private, recreational, or sport pilot certificate.  The PIC must also hold a current FAA airman medical certificate or a valid U.S. driver’s license issued by a state, the District of Columbia, Puerto Rico, a territory, a possession, or the Federal government.  The PIC must also meet the flight review requirements specified in 14 CFR §61.56 in an aircraft in which the PIC is rated on his or her pilot certificate.


So, the FAA could decide to issue an exemption under section 333 for such a flight, and declare whatever limitations they wanted (or, of course, reject it outright).  If approved, I'd expect it to come with limitations like:


Must adhere to 14 CFR 107, treating the flight as an sUAS flight even though it exceeds the sUAS maximum weight.
The PIC must be certificated for Remote Pilot (sUAS) or hold an airline transport or commercial certificate with a corresponding class 2 medical and valid government-issued photo ID.


(Every other restriction I could think of is already part of 14 CFR 107, and the carried human would probably be treated equal to a wing walker... and I can't dig up any extra restrictions regarding them)