Does this verse say that a believer can be kicked out of the covenant?
I assume you’re thinking of Matt 18:15-17 (NIV) 
I would suggest that the aim is to “win them over”, so “treating them as you would a pagan”, amongst other things, must at least involve retelling them the Gospel, motivated by love. They might not react well and leave that church, but it doesn’t mean God has given up on them.
St. Paul’s grafting analogies in Rom 11, as well as where he arrives on the subject, are relevant to that GosMatt verse, too. Being kicked out of the covenant (or kicking one’s self out, either one, although the thrust of action in both cases is being kicked out) isn’t hopeless.
How are we to understand that another believer has reached this point? When they won’t repent? Aren’t we all unrepentant at some level, though?
I would also add relavence Galatians 6:1:
*“Brethren, if a man be overtaken in a fault, ye which are spiritual, restore such an one in the spirit of meekness; considering thyself, lest thou also be tempted.” *
This is the proper follow-up in the case of one in the Matt. 18. We have the example of I Cor. 5, where one is having unnatural relationship with his father’s wife (not sure if that means his own mother or another wife that his father married later). At any rate, the injunction “that he that hath done this deed might be taken away from among you” and
“deliver such an one unto Satan for the destruction of the flesh, that the spirit may be saved in the day of the Lord Jesus” is part of the discipline that a church needs to follow to ensure that the sin is not spread within the camp. Unfortunately, we have had similiar instances in our church where the Pastor was forced to let the person no longer be part of our fellowship because of the effect it might have on other members, especially the young people. It may sound intolerant, but if there is no church discipline, then it’s very easy for a church to fall into complacency. A couple who makes ends up pregnant and refuses to marry sends a wrong message. However, I’ve also seen where that kind of mistake happened and the couple repented and gets right with the Lord by getting married as soon as they can. They remain active members of our church still.
As far as “delivering up to Satan”, we see an example of this in I Timothy 1:19-20,
“Holding faith, and a good conscience; which some having put away concerning faith have made shipwreck: Of whom is Hymenaeus and Alexander; whom I have delivered unto Satan, that they may learn not to blaspheme.”
Deliver means “to hand over” or “commit”. It has the connotation that the person is on his/her own, without the benefit of fellowship in the church. I don’t think that means that he/she is no longer a Christian, nor necessarily excommunicated from the church, but that just as a parent might kick their child out of the house, (not because they don’t love them, but ‘for their own good’), so as to find out how hard it is in the world to live without parental guidance or godly influence, with the hope that the son or daughter will realize their faults and seek reconciliation, not unlike story of the the Prodigal Son.
So there is always hope for repentance, even for the most egregious offense, as we are taught to forgive all.
And BTW, the fault of Hymenaeus was apparently one of heresy, according to 2 Tim. 2:16-18, which would be conflictive to the sound doctrine of the church. In one Baptist churh where I was a member, we had one teaching Calvinism in a Bible study that he was conducting in his home with several church members present. He was confronted several times, by a deacon, then the Pastor and the same deacon, then finally was told in front of the congregation of his refusal to stop and consequently was asked to leave.
Same. It’s usually unpleasant but I understand the need to care for the flock.