That would depend in your citizenship, which you haven't provided.
The Faroe Islands is not part of the Schengen Area.
The rules that would otherwise apply to Schengen Area residence permit/card holders, does not apply for the Faroe Islands.
If your citizenship normaly requires a visa to enter the Schengen Area, then you must apply for a visa that is specifically valid for the Faroe Islands.
@phoog: More specific information about travel to the Faroe Islands with a Schengen residence permit, including information about how to apply for an "entry permit," is available at:
Visa to the Faroe Island or Greenland - How to apply
Passport & visas - Visit Faroe Islands
B. EU-COUNTRY AND SCHENGEN COUNTRY
You do not need a visa or other permit to visit the Faroe Islands. You may enter with a passport, or, for visits, a national ID card (that includes a photo) from one of the below countries. This only applies to citizens of the below countries.
It is NOT sufficient to be a foreign resident; you must be a citizen.
C. OTHER COUNTRIES
The Faroe Islands are NOT part of the Schengen Area. Consequently, people cannot enter the Faroe Islands based on their Schengen visas or Danish visa. When a visa is applied for at the Danish Embassy, it must be specifically for the Faroe Islands.