It may be more accurate to ask what was the Corinthians' problem with speaking in tongues and what was Paul's admonition regarding speaking in tongues. Paul states that when speaking in tongues, the person is speaking to God--now I don't think he is inferring that this is a problem, do you? Paul delimits the value to the Church of speaking in tongues by saying that since one who speaks in tongues does "utter mysteries by the Spirit," the meaning of which are unknown to the brothers and sisters in the Church, speaking in tongues must always be accompanied by the interpretation of prophesy--or--the value of tongues to the Church is purely ornamental and not practical nor useful.
Paul further admonishes that the value to nonbelievers of speaking in tongues is even more limited since, if a nonbeliever were to enter when a congregation was collectively speaking in tongues, "will they not say that you are out of your mind?" From this we can infer that the Corinthians' problem with speaking in tongues was that they, as with other aspects of their Christian lives, were putting personal experience ahead of spiritual reality--choosing what felt good or right above what was good or right. Paul's admonition to the Corinthians was to remember the reverential, personal, worshipful nature of the gift of speaking in tongues and to use it appropriately while valuing the gift of prophesy higher for its greater value in public gatherings, both for believers and nonbelievers.
Paul points towards the way that the gift of tongues was actually causing dissent and problems within the church rather than being used to unify Christians and glorify God. It is vitally important to remember that the two chapters that talk about spiritual gifts and how they should be used and administered properly in a church setting (1 Corinthians 12 and 14) are separated by one of the most famous passages in the Bible: 1 Corinthians 13, that talks about the true nature of love. The meaning is obvious: Paul is concerned that spiritual gifts, including the gift of tongues, were not being used in love and to build up the church.
In chapter 14 of I Corinthians, Paul believed that it is better to prophesy than to speak in an unknown tongue. It stated that speaking in an unknown tongue is a prayer language to God, but it does not edify or build up the body of Christ or the church.
2For he that speaketh in an unknown tongue speaketh not unto men, but unto God: for no man understandeth him; howbeit in the spirit he speaketh mysteries.
He felt that those who spoke in an unknown tongue did not help the church. He felt that those who spoke in an unknown tongue should prophesy or translate in order to edify and build up the church people. He said that it is greater to prophesy because everyone will understand what was spoken:
5I would that ye all spake with tongues but rather that ye prophesied: for greater is he that prophesieth than he that speaketh with tongues, except he interpret, that the church may receive edifying.
The problem that Paul had was the fact that some were speaking in tongues and not interpreting for the edifying or building up of the church. He felt it would cause confusion and that there would be no purpose:
6Now, brethren, if I come unto you speaking with tongues, what shall I profit you, except I shall speak to you either by revelation, or by knowledge, or by prophesying, or by doctrine?
Paul desired that there be clarity when speaking in tongues.
According to Paul, the problem with speaking in tongues is that it does not do anything for the church as a whole. It does not help anyone other than the person who is doing it.
To Paul, the point of prophecy and speaking out is to help other people. It is to help them understand more about God and about what God wants from them. Regular prophecy does this, he says, but speaking in tongues does not. Paul seems to be saying that speaking in tongues is a self-indulgent act that is meant more to make themselves look good rather than to help others.
At the beginning of I Corinthians 14, Paul states in several different ways that speaking in tongues is primarily a self-pleasing act. In verse 2, Paul describes speaking in an unknown tongue as speaking to God but not to man. While this in not a negative act, Paul would rather humans be focused on "edifying" (building up/helping) one another than building up themselves in front of others.
Similarly, in verses 9 and 11, Paul stresses that if someone speaks in an unknown tongue, the listener cannot be helped because he does not have a clue what the speaker is saying. Paul essentially questions the purpose then of the speaker using an unknown tongue. Verse 11 goes on to say that the practice not only does not help the non-comprehending listener but that it also hurts him because he questions his intelligence and even worse--his spirituality.