This is the conclusion of what I began in The Missionary Use of Tongues: A Snipe Hunt.
1 Corinthians 14 makes an explicit comparison between prophecy and tongues, with Paul clearly preferring prophecy. This much any cessationist will tell you. However, it is instructive to recognize in what context and for what reasons Paul prefers prophecy. Paul's argument throughout is that prophecy is preferable specifically because it is intelligible to others. In other words, Paul is discussing what is appropriate within the context of corporate worship. He really hasn't departed from his overall topic of what is appropriate in worship since chapter 11. What Paul says about prophecy in chapter 14 is that the person who prophesies "speaks to men for their strengthening, encouragement, and comfort" (v. 3), "edifies the church" (v. 4), that only "two or three prophets should speak" (v. 29), that those who prophesy should do so "in turn so that everyone may be instructed and encouraged" (v. 31), that "the spirits of prophets are subject to the control of prophets" (v. 32), that a genuine prophet would acknowledge that what Paul is writing "is the Lord's command" (v. 37), and that people should "be eager to prophesy" (v. 39). All of this is clearly within the context of corporate worship.
Tongues is here being denigrated by comparison, but it is being denigrated specifically because it is unintelligible. This is an odd thing for Paul to assert if tongues was given for the purpose of communicating the gospel to people in other languages. It is true that the Corinthians were evidently misusing the gift, but if their misuse consisted in using the gift in a context in which there are no foreign-language listeners, Paul never says so. He also never says that they are using a counterfeit gift. He never tries to stop them from using the gift. Paul's argument against tongues takes an entirely different line.
Tongues in Private and in Public
Paul's essential argument regarding the use of tongues is that it is personal and unintelligible to others; therefore it is unsuited to corporate worship unless it is accompanied by interpretation. Paul acknowledges that speaking in tongues builds up the speaker, and says that he would like all of them to speak in tongues (vv. 4-5). This is hardly a carte blanche put-down of tongues. However, in the church, it is better to prophesy, because corporate worship is about building up the body together, not individuals separately. What goes on in the service must be intelligible, so that others may receive benefit. Paul is grateful that he speaks in tongues more than all of them, but "in the church" he would rather "speak five intelligible words to instruct others" (vv. 18-19).
To say that it is unsuited for corporate worship is quite different than to say that it is unimportant in itself or false altogether. What we see here is the dichotomy between tongues being used for personal devotion (yes, a "private prayer language") and the far more restricted use of tongues in corporate worship, in which interpretation is insisted upon (vv. 13, 27-28). It is quite interesting that despite Paul's preference for prophecy over tongues in corporate worship, he doesn't ban tongues altogether (v. 39) or even banish their use to the private prayer closet. He gives guidance on the use of both tongues and prophecy, based on the underlying principle that "all of these must be done for the strengthening of the church" (v. 26).
Incidentally, verse 14 contradicts another frequent allegation, that Paul's references to his own speaking in tongues (cf. v. 18) merely refer to his own naturally learned ability to speak in a number of different languages. Paul specifically says that if he prays in a tongue, his "mind is unfruitful." This is obviously not a reference to speaking in a naturally learned second language; moreover, the context involves prayer, and it doesn't make much sense to imagine that Paul would make a point to pray in a naturally known second language.
So in the end, the entire context of 1 Corinthians 14 contradicts the supposition that tongues was intended solely, or even primarily, for the purpose of facilitating missionary activity. This, in and of itself, has no bearing on the cessetion debate, but it does remove the objection that the purpose of tongues as described in the Bible is significantly different than the use of tongues in modern Pentecostal and charismatic circles. Responsible Pentecostals and charismatics have always insisted upon the strictures of 1 Corinthians 14 being followed, and the strictures that many cessationists have placed upon tongues do not bear scriptural scrutiny.
 Although Paul's primarily focus is on what will be heard by others in corporate worship, he does allow a measure of quiet, personal tongues in a worship service in verse 28.