Cape Lookout does.
From the regs: Open campfires can only be built below the high tide line. Visitors may want to check the tide predictions in advance to determine when they can build fires. Loose dead wood on the ground within 200 feet of the shore may be used for campfires. Wood from shipwrecks or historical or archeological sites and wood from standing trees may not be used. Campers are encouraged to bring their own wood because the amount of wood available in the park is extremely limited.