Does a church member have the right to privately confront or send a message to an elder, for the purpose of disagreement or criticism? If this not acceptable, what consequences should follow a disagreement?
"Obey those who rule over you, and be submissive, for they watch out for your souls, as those who must give account. Let them do so with joy and not with grief, for that would be unprofitable for you" (Hebrews 13:17).
Elders are fellow members of the church. As men, they have a tendency to make mistakes like everyone else. They have been given the duties of an elder because they have managed to live an exemplary life and their advice and guidance is valuable to the congregation.
Because of the nature of their work, they have a difficult time. People are watching, so you must be particularly watchful against sin in your own life. Sadly, the elder tends to also see the worse. He's called in to handle the problems and one can become jaded when all you see are problems. And then there are the members who mistakenly think that elders have placed themselves on pedestals and it is their job to shoot them down. That is why the Hebrew write warned the people benefiting from the oversight of elders not to make their lives miserable. It is a hard enough task as it is.
If you see a problem, then by all means approach one of the elders and point out the difficulty. If the elder is caught up in sin, and you can't get him to see the problem, then talk to one of the other elders or the preacher. They will rightly ask for evidence, "Do not receive an accusation against an elder except from two or three witnesses. Those who are sinning rebuke in the presence of all, that the rest also may fear" (I Timothy 5:19-20).
If it is a matter that you don't agree with how an elder is handling a problem, then ask him why he is choosing this path. You never know, you might learn something. However, it would be foolish to tell an elder how to do his job when there is a reason he is in the position and you are not. It would be equal to telling your state senator how he ought to be conducting his campaign for re-election. Yes, you might see something that he has overlooked, but also realize that he has far more experience, is privy to far more information than you, and might have very good reasons for the choices he has had to make.
"Likewise you younger people, submit yourselves to your elders. Yes, all of you be submissive to one another, and be clothed with humility, for "God resists the proud, but gives grace to the humble."" (I Peter 5:5).