God is very serious about the governance of his Church. Biblical discipline (Matt. 18:15-18) is applicable to all, including its leadership, i.e., elders. Both false teaching and unholy living could be in view here.
Lea points out:
In 1 Timothy 5:19-20 Paul discussed the process of discipline for erring leaders. First, he warned against accepting an accusation against an elder unless two or three witnesses support it. Paul was not urging special treatment for the elder, but he was urging fair protection from capricious accusations. The Church leader should enjoy at least as much protection as the ordinary Jew had under the law (see Deut 17:6; 19:15).
"Elect angels" refers to those who will presumably serve as witnesses at the judgment (cf. Matt. 25:31; Rev. 14:10). See WCF 3.3; WLC 13, 19. Hughes categorizes this series of verses into three headings: (1) caution, (2) courage, and (3) fairness:
Caution. Paul begins with a cautionary note: "Do not entertain an accusation against an elder unless it is brought by two or three witnesses" (1 Tim. 5:19). A high degree of caution is in order because pastoral leadership is a profession that depends on character. If you do not maintain character, you can lose everything except salvation. Next to your life in Christ, your character is your most valuable possession. . . . [Note], Church leaders are highly visible and are tragically vulnerable to the adverse actions of the disorderly, the malevolent, and the ill-willed - and to whispering gossip.
Courage. Though we must be cautious in accusing, if the charge is substantiated, we must have the courage to rebuke: "Those who sin are to be rebuked publicly, so that the others may take warning" (1 Tim. 5:20). This may sound cold and unloving, but it must be done, for the sake of the church "so that the others may take warning" - both minister and congregation. This is where today's church has lost its nerve. Leaders sin with impunity - and then move on to other churches to do the same thing. We must determine not to fall to such a loss of courage but rather to lovingly confront those who are doing wrong. A lack of fortitude is not loving but unloving - unloving of Christ, unloving of the church, and unloving of the offender.
Fairness. Lastly, in this matter of discipline there is the issue of fairness. 1 Tim. 5:21 is closely joined with 1 Tim. 5:19,-20, and the NIV's beginning a new paragraph here is "a mystery," as Gordon Fee says. Its connection is urgent. "I charge you, in the sight of God and Christ Jesus and the elect angels, to keep these instructions without partiality, and to do nothing out of favoritism" (1 Tim. 5:21). Paul's force and passion here must be because there had been a scandalous exercise of favoritism shown in reference to some sinning church leaders. The Ephesian "old boys" system had done its thing. Some guilty leader had gotten away with murder, so to speak.
It is unfortunate today that many denominations go overboard in protecting their elders/pastors. Some will not even accept an accusation against one no matter how well documented it is. They chose to overlook their false doctrine if enough money keeps rolling in to the denomination. This said, some denominations are too hard on elders accepting and acting upon false accusations. Some see Church discipline (Matt 18:15-20) as a way to rid themselves of a alleged problem instead of restoring said one to Christ (Gal 6:1-10).
two or three witnesses. Those who teach are also accountable to the covenant community they serve. 'False teachers' (see note 1:3) or those who are otherwise not 'above reproach' (see note 3:2) can be disciplined, even removed from leadership on the basis of sworn testimony from 'two or three witnesses.' As Jesus is reported to have done in Matthew 18:15-17, Paul appealed to Deut 19:15, which underscored the seriousness of the charge and its corresponding sanction by requiring more than one witness.