In 1 Cor 6:5, Paul talks about shaming the Corinthians, clearly being astounded at their immaturity and lack of Biblical wisdom. Earlier (1 Cor 4:14), he has written not to shame but to warn them, but now he wants them to be ashamed and to change their ways as a result. Shame (a painful feeling of humiliation or distress caused by the consciousness of wrong or foolish behaviour) can be a powerful motivator to godliness, but the point of shaming someone in a way that is approved by God is to make them see the error of their ways and offer them the opportunity to repent and change. It is not an end in itself. Just as Paul’s intention behind expelling the immoral brother in 1 Cor 5 was ultimately salvation and restoration (1 Cor 5:5, ‘so his spirit may be saved on the day of the Lord’ and 2 Cor 2:7, ‘you ought to forgive and comfort him, so that he will not be overwhelmed by excessive sorrow’), his intention in this passage is to get the Corinthians to change how they are settling disputes amongst themselves.
Shame can be, in Robert Douglas-Fairhurst’s words, ‘a powerful social corrective.’ (http://www.telegraph.co.uk/culture/books/bookreviews/11376490/Is-Shame-Necessary-New-Uses-for-an-Old-Tool-by-Jennifer-Jacquet.html) Nonetheless, we do well to realise that some people are over-sensitive to sin and guilt and can become very easily condemned by others, feeling ashamed when there may well be no legitimate reason to be ashamed. The experience of shame can help to reinforce social norms within an organisation and so help social cohesion, but the reason the Corinthians needed to change was to live out their identity as children of God. A desire to live as God wants because we long to please Him seems to me to be better reason for holiness than simply wishing to avoid other people’s condemnation or value-judgments.