Shame is a sense of guilt, disgrace, loss of favor with God. Shame is something that Christians aren't to have, Christ took away our shame at the cross and He makes us righteous. Shame is also not to be used to manipulate people, it is not to be a motivator. But there are two places in the New Testament where Paul seems to shame people.
1 Corinthians 6:1-6
"Does any one of you, when he has a case against his neighbor, dare to go to law before the unrighteous and not before the saints? Or do you not know that the saints will judge the world? If the world is judged by you, are you not competent to constitute the smallest law courts? Do you not know that we will judge angels? How much more matters of this life? So if you have law courts dealing with matters of this life, do you appoint them as judges who are of no account in the assembly? I say this to your shame. Is it so, that there is not among you one wise man who will be able to decide between his brethren, but brother goes to law with brother, and that before unbelievers?"
The word translated shame here is defined as "a turning in upon oneself." It is not the usual Greek word translated "shame."
In 1 Corinthians 6:5 the issue is that Christians are turning to non-Christians or so-called brothers who are immoral to settle their problems for them. Understanding the correct definition of the word we can easily see Paul saying, "I say this so you will turn in upon yourselves and look for some wise man who will be able to decide between his brethren." Paul is telling them to look at themselves and amongst themselves to find a wise man to judge their matters. Paul is not shaming them with guilt. It is the opposite, the normal word for shame means disgrace but Paul encourages their competence as saints (which they are by grace) and communicates, "Look at yourself. Are you able to help settle these matters?"
1 Corinthians 15:34
"Become sober-minded as you ought, and sin not; for some have no knowledge of God. I speak this to your shame."
The word translated shame here is the same as the one above and is defined as "a turning in upon oneself." These are the only two places this word appears in the New Testament. Paul tells them to "Wake up, become sober-minded." Understanding the correct definition of the word we can see that Paul is saying "Wake up, take a look at yourselves!" The NLT puts it this way "To your shame I say that some of you don't know God at all." Meaning "look inwardly, examine yourselves to see if you know God." This scripture is about realization not shame and guilt. Even if we took the word shame here to mean guilt or disgrace, which it doesn't, it is still aimed at non-Christians and Paul is not shaming Christians. "To your shame I say that some of you don't know God at all." Meaning "To the some who don't know God I appeal to the presence of your shame to help you realize that you don't know God."
Judges in the Assembly Part 2: A closer look at the context & intent of 1 Corinthians 6