That's one of several important questions John MacArthur answers in the booklet Answering the Key Questions About Elders.
Here's what he says:
Elders are called and appointed by God, confirmed by the church leadership, and ordained to the task of leadership. To them are committed the responsibilities of being examples to the flock, giving the church direction, teaching the people, and leading the congregation. Scripture implies that anyone at a lower level of leadership should be under the elders' authority.
Because they share unique responsibility and position in the church, elders are worthy of great respect.
First Thessalonians 5:12-13 says, "We request of you, brethren, that you appreciate those who diligently labor among you, and have charge over you in the Lord and give you instruction, and that you esteem them very highly in love because of their work."
The Greek word translated "appreciate" in that passage means, "to know intimately." Along with the rest of this passage, it implies a close relationship involving appreciation, respect, love, and cooperation. That great feeling of appreciation is to arise "because of their work." We are to respect them because of the calling they are fulfilling--not only because of their diligent labor and the task they have but because that calling is so noble.
Hebrews 13:7 says, "Remember those who led you, who spoke the Word of God to you; and considering the result of their conduct, imitate their faith." That emphasizes both the elder's responsibility to live as an example, manifesting virtue in his life, and the church's duty to follow their example.
Verse 17 adds another dimension of the congregation's duty toward their spiritual leaders: "Obey your leaders, and submit to them; for they keep watch over your souls, as those who will give an account. Let them do this with joy and not with grief, for this would be unprofitable for you." In other words, the congregation is spiritually accountable to the elders, and the elders are accountable to God. The congregation should submit to the elders' leadership and let the elders be concerned with their own accountability before the Lord. And if the congregation is submissive and obedient, the elders will be able to lead with joy and not with grief, which is ultimately unprofitable for everyone.
That does not mean, however, that if an elder sins openly his sin should be ignored. First Timothy 5:19-21 says, "Do not receive an accusation against an elder except on the basis of two or three witnesses. Those who continue in sin, rebuke in the presence of all, so that the rest also may be fearful of sinning. I solemnly charge you in the presence of God and of Christ Jesus and of His chosen angels, to maintain these principles without bias, doing nothing in a spirit of partiality.
An accusation of sin against an elder is not to be received lightly. Nor is it to be overlooked. Elders are to be disciplined for sinning in the same way anyone else in the church would be. In no way are they to receive preferential treatment.
The testimony of the church is most visible in the lives of the elders. If they ignore the biblical mandate for holiness, the church will suffer the consequences. Equally, if the church is not submissive to the leadership God has ordained, its testimony will suffer, its effectiveness will be diminished, its priorities will be unbalanced, and ultimately its flavor as the salt of the earth will be lost.
©1984 by John MacArthur. All rights reserved. Unless otherwise identified, all Scripture quotations are from the New American Standard Bible, ©1960, 1962, 1963, 1968, 1971, 1973, 1975, 1977, and 1995 by The Lockman Foundation, and are used by permission.