Obadiah is not quoted, or even alluded to, in the New Testament. Nonetheless there are still a few points of contact worth noting. Herod the Great was an Idumean, that is, an Edomite. His attempt to have Jesus killed (Mt. 2:16) is of a piece with Edom’s historic hostility to God’s people.[1]
Positively, Mark 3:7-8 includes people from Idumea among those who followed Jesus’s ministry. If these people came to faith in Christ, they would be among those resurrected believing Edomites who would be ruled by Yhwh from Jerusalem during the millennium and new creation.
Finally, the concept of the Day of the Lord, introduced primarily by Obadiah, is prominent in the Thessalonian epistles and in Revelation. Obadiah warned of the nations drinking the cup of Yhwh’s judgment and Revelation 14:10 speaks of those who will “drink the wine of God’s wrath” during that final day of the Lord.
[1] Rooker, “The Book of Obadiah,” in The Word and the World, 444.