Permafrost did not prevent mining in the Yukon, or diamond mining in Siberia. There was also the reality TV show Goldrush Alaska.and also a rather large oil pipeline in Alaska built on permafrost.
At present, there are six exploitation licences in Greenland: a gold mine in south Greenland, an iron ore deposit, a gemstone deposit, a lead and zinc mine and an anorthosite mine in west Greenland, and a lead and zinc mine in northern Greenland. Greenland Minerals and Energy, is developing a rare-earth mine at Kvanefjeld which will produce uranium, fluoride and thorium as well as rare-earth minerals.