Obviously we could do that. At some (high) price it would be profitable to extract and export natgas and shale oil.
But that would make our energy prices much higher as well.
Oil prices vary only slightly around the world based on transportation cost, so oil prices would not be affected by permission to export oil. Natural gas prices would not be affected much by large LNG export volumes. If prices rose $1.00/mcf, about 20%, known plays would become economic and production would increase sharply. Meanwhile, exploration and production and LNG facilities construction would boost the U.S. economy.