Here are a few comments from some commentaries on the phrase, "those who do not know God and those who do not obey the good news about our Lord Jesus" (2 Thess 1:8 NWT):
This one is from the NICNT-Thessalonians (Leon Morris, p.203-4):
Bearing in mind the Hebraistic coloring of this passage and the fondness for parallelism in elevated Hebrew style, we apparently have here two designations of more or less the same group of people. There is nothing in the context to prepare us for an allusion to Jew or gentile. "Those who do not know God" refers, of course, not to peopel who have never heard of the true God, but to those who are culpably ignorant. It is the sort of thing that Paul speaks of in Romans 1:28, where he refers to people who "did not think it worthwhile to retain the knowlledge of God." The second clause is then a specific example of this, and the most heinous of all, for it invokes the rejection of the revelation that God has given in his Son. The gospel is a message of good news, but it is also an invitation from the King of kings. Rejection of the gospel accordingly is disobedience to a royal invitation. This is emphasized in the reference to Jesus as "Lord," a word that, in addition, has point in a second advent setting.
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This quote is from the NAC-Thessalonians (D. Michael martin, p.212):
Both clauses begin with the definite article, and such a construction would normally indicate that the author was thinking of two distinct groups. The two groups could be interpreted as the Gentiles who do not know God and the Jews who know God but are disobedient to the gospel. Although this is possible, the Greek construction does not require it, and the distinction is misleading. "Know" in this context refers to obedience as much as knowledge, and given this sense of the word both Gentiles and Jews may not "know God" (see John 8:54-55; Jer 9:6). Both unbelieving Jews and unbelieving Gentiles were willfully disobedient to the gospel as far as Paul was concerned (Rom 1:18-20; 2:12-16). Thus either a Jew or a Gentile could be described as one who does not know and does not obey God. Rather than presenting two groups, these two parallel clauses probably reflect the Old Testament background of the passage and utilize parallelism to describe a single group: unbelievers of any ethnicity.
In support of both clauses referring to a single group it should also be remembered that both clauses apparently describe those who are troubling the church (v. 7). Thus the words "do not know God" could not apply to people who are ignorant of God and/or of the gospel but only to those who have rejected them. 32 The second clause in the parallel, "and do not obey the gospel of our Lord Jesus," adds emphasis and at the same time brings back into focus the thrust of the passage. God's judgment will fall on those who neither acknowledge God nor obey the gospel, and the ultimate expression of such rejection is the persecution of the church (cf. 1 Thess 2:14-16). Thus condemnation is earned not by ignorance or by an isolated act of persecution of believers but by the settled rejection of God inspiring the persistent persecution of God's people.
Footnote 32 reads:
The fate of persons who have never heard the gospel is not discussed in this passage (cf. Rom 1:18-32).
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Since "everlasting destruction" is involved in the passage, it is interesting to compare this with some research I posted here on "unforgivable sin." See my post # 682, 683)