They didn't leave many genetic traces but there is some Paleo-Eskimo mtDNA in the modern indigenous North American gene pool. As usual, this leaves the paradigm of North American population history intact.
The introduction to the paper explains:
The Beothuk were members of the “Red Ochre” culture (Marshall 1996) that inhabited the island of Newfoundland on the extreme Atlantic coast of North America ca. 2.1–0.2 ka BP, at the end contemporaneously with an extant Algonquian people, the Mi’kmaq, prior to and during early European colonization of the interior <1800 CE. The Beothuk are generally accepted to have become culturally extinct with the deaths of the last known members, Shanawdithit (1801?–1829), her uncle Nonosabasut (?–1819), and his wife Demasduit (?–1820) (Marshall 1996).
The Beothuk succeeded the immediately previous inhabitants of the island and (or) the adjacent coast of Labrador, the Maritime Archaic ca. 8.0–3.4 ka BP, after an interval of ca. 1.4 ka (Renouf and Bell 2011). Other peoples in this area at this time include the Dorset Paleo-Eskimo, who overlap temporally with both the Maritime Archaic and Beothuk ca. 2.1–1.1 ka BP (Renouf 2011). An outstanding question is the possible survival of Beothuk genetic lineages in the modern Mi’kmaq and (or) other modern persons of Native American descent (Aylward and Joe 2018).
The paper and its abstract are as follows:Duggan et al. (2017) recovered mitogenome sequences from aDNA of individuals representative of the Maritime Archaic, Dorset, and Beothuk peoples, on the island of Newfoundland and adjacent southern shore of Labrador, dating from ca. 7700 BCE – 1820 CE. Inter alia, they concluded that there is a genetic as well as temporal discontinuity between the Maritime Archaic and the Beothuk, and that the former are not the ancestors of the latter. Here, Beothuk and Maritime Archaic mitogenomes are compared with the most closely related sequences of modern persons in the GenBank database. These comparisons bear on the utility of mtDNA sequence data for the association of living persons to extant or extinct Native American peoples.
The Beothuk were a Native American people who formerly occupied the island of Newfoundland, and who are generally accepted to have become culturally extinct in 1829. The Beothuk succeeded the Maritime Archaic people on the island after a hiatus of ca. 1.4 ka, and were themselves succeeded by the extant Mi’kmaq within historic times. Genetic continuity between ancient and modern Native Americans remains of interest.
Complete aDNA mitogenomes from ancient Beothuk and Maritime Archaic were compared with the most closely related modern mitogenomes as obtained by BLAST search of GenBank.
Beothuk mitogenomes in five clades are in one case identical to and otherwise differ by minima of three to eight SNPs from the most closely related modern mitogenomes.
Maritime Archaic mitogenomes in 12 clades are in one case identical to and otherwise differ by minima of one to nine SNPs from the most similar modern mitogenomes.
The single available modern Mi’kmaq mitogenome differs from the most similar Beothuk and Maritime Archaic mitogenomes by 12 and 22 SNPs, respectively.
Phylogenetic analysis and sequence similarities imply lineage extinction of most ancient clades, as well as continuity of two Beothuk and at least one Maritime Archaic lineages in modern Native Americans and their descendants.
Steven M. Carr, "Evidence for the persistence of ancient Beothuk and Maritime Archaic mitochondrial DNA genome lineages among modern Native American peoples" Genome (April 13, 2020) https://doi.org/10.1139/gen-2019-0149 (open access).