According to the Mineral Resources Act section 45, permanent residents of Greenland may collect and extract minerals without a licence, if the value of the minerals do not exceed DKK 100.000 pr. year.
Tourists are not allowed to collect or extract stones/mineral resources in Greenland. However, tourists can buy stones/mineral resources from local craftsmen, small scale licensees or shops in Greenland. Remember to bring your receipt with the stones/mineral resources when leaving Greenland. You can find more information on export here.