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The theme for verses 17 and 18 is found in Malachi 1:2-3:
"I have loved you," says the LORD. "Yet you say, 'In what way have You loved us?' Was not Esau Jacob's brother? " says the LORD. "Yet Jacob I have loved; but Esau I have hated, and laid waste his mountains and his heritage for the jackals of the wilderness."
God's choice is supreme. He made His sovereign choice of Jacob over Esau before either had done anything. They may have struggled in the womb, but He made His choice prior to their developing any character. He chose Jacob, and that is the end of the matter.
Obadiah writes in verse 17, "But on Mount Zion there shall be deliverance, and there shall be holiness; the house of Jacob shall possess their possessions." This verse introduces an interesting distinction: "On Mount Zion [is] deliverance," but the end of verse 18 says, "No survivor shall remain of the house of Esau." The destinies of these two peoples are total opposites. Whereas God loves Jacob and allows a remnant to survive into the Millennium, no one survives of Esau.
There is no way to know how absolute this pronouncement may be. Will there be, perhaps, some few Edomite survivors counted among those who are converted—who become part of spiritual Israel, in effect? Perhaps, but certainly all the proud and gloating of Edom will be completely annihilated.
Verse 18 tells us, "The house of Jacob shall be a fire, and the house of Joseph a flame." "The house of Jacob" may refer particularly to Judah, and "the house of Joseph" would then refer to the rest of the nations of Israel, led by the Joseph tribes, Ephraim and Manasseh. In any case, it indicates the entirety of Israel. Zechariah 12:6 contains similar language, in which the governors of Judah will be "like a firepan in the woodpile" and "shall devour all the surrounding peoples." Edom will be one of these devoured nations.
Richard T. Ritenbaugh
All About Edom (Part Five): Obadiah and God's Judgment