On 1 July 2024, the Greenland Parliament Act on local mineral activities came into force. It follows from section 9(4) of the act that a non-permanent resident (i.e. who has not had residence and tax liability in Greenland for the past 5 years) no longer needs an export permit to export minerals that have been collected or quarried for scientific purposes. In other words: You no longer need an export permit if you, for example, collect a rock sample in connection with a scientific activity for which the Expedition Office issues a permit.
However, the exception only applies if the collection is carried out as part of the scientific activity. There is no free export for purposes other than scientific.
Please see Mineral Resources Authority - Naalakkersuisut | Greenland or reach out to mlsa@nanoq.gl for guidance on any specific case.