Do the gifts of prophecy and tongues continue today? Before I answer this, I want to answer a more narrow question: Are all of the spiritual gifts available to every Christian? In 1 Corinthians 12:29-30, Paul asks a series of rhetorical questions:
All are not apostles, are they? All are not prophets, are they? All are not teachers, are they? All are not workers of miracles, are they? All do not have gifts of healings, do they? All do not speak with tongues, do they? All do not interpret, do they?
Paul's position shows that not all gifts are available to everyone. I’m urging this in support of the concept of offices, or the idea that God selected certain individuals for certain functions in the Church. We also see this in Ephesians 4: “[Christ] gave some as apostles, and some as prophets.” These gifts are not distributed equally. They are not part of the broader inheritance that all Christians receive from Christ.
Now as to whether the gifts of prophecy and of tongues continue to the present day, let's look at 1 Corinthians 13. Here, Paul lays out the role and function of the gifts in the Church, stating they will cease when "perfection" comes. The most obvious sense of "perfection" is the eschaton, or end of times. Many charismatics take this to indicate that the gifts of tongues and prophecy will continue to the end of time, and so would I, if it were not for a parallel but more detailed discussion in Ephesians.
In Ephesians 2:20 and 4:11-16, we have an examination of the role and duration of the gifts, a parallel to the examination Paul gives in 1 Corinthians 13-14, using very similar language, but with added detail:
And he gave the apostles, the prophets, the evangelists, the shepherds and teachers, to equip the saints for the work of ministry, for building up the body of Christ, until we all attain to the unity of the faith and of the knowledge of the Son of God, to mature manhood, to the measure of the stature of the fullness of Christ.
The Apostles are included in this account of gifts ("He gave..."). The Apostolic office has surely ceased. Neither were all Christians without discrimination Apostles in the beginning of the Church, nor is anyone today an Apostle ("Last of all, as to one untimely born, he appeared also to me. For I am the least of the apostles, unworthy to be called an apostle"). There is, then, one office/gift that is lumped together with the other gifts (that are given to build up the church till the eschaton), yet upon further examination we can say is both limited in scope and duration.
Now turn to Ephesians 2:20: "[the Church] built on the foundation of the apostles and prophets, Christ Jesus himself being the cornerstone." Many on a skim reading take the "prophets" in this passage to be referring to the Old Testament prophets. However, the passage refers to the New Testament prophets (see Ephesians 3:5). The Apostles (whose office has surely ceased) and New Testament prophets are both included as the foundation laying office. The Greek use of the word for "foundation" indicates a completed action. (The other offices, such as teachers, evangelists, etc., are not included in the "foundation" laying, and will continue in duration till the end of time.)
So, we've seen that the gifts are not given indiscriminately to all (1 Corinthians 12:29-30), that some offices are limited in duration (1 Corinthians 15:8-9a), and that the prophets are classified alongside the Apostles as a foundation laying office (Ephesians 2:20). The clear cessation of Apostles, the linking of prophets with Apostles, and the use of the word "foundation" in reference to both Apostles and prophets indicates cessation of the prophetic office. The one account that seems to implicitly indicate that all gifts will continue till the end, 1 Corinthians 13, has a parallel account in Ephesians 2-4, in which Paul discusses the gifts individually and with greater detail, namely, the "foundation" detail--so Paul moved from a general account of the gifts in 1 Corinthians 13, to a more particular, detailed account in Ephesians 2-4. Ephesians 2-4 should control our discussion of the particular gifts, as it is where Paul comments on the gifts particularly.
What about tongues?
The final step in this argument is the equivalence Paul makes between the gift of tongues interpreted and the gift of prophecy: "I wish all of you spoke in other languages, but even more that you prophesied. The person who prophesies is greater than the person who speaks in languages, unless he interprets so that the church may be built up." - 1 Cor. 15:5
Tongues (speaking in "other languages") becomes equivalent to prophecy (see also Acts 2:15-18 for this equivalence) when understood by others. It is inferior (and prohibited even in the foundation-laying era of the Church) insofar as it is not interpreted. So, interpreted tongues is equivalent to prophecy, which has ceased as the Apostolic office has ceased.
Or a Cessationist could take their Cessationism as a sort of working-hypothesis: "I've not seen a genuine manifestation of the gifts, so they must have ceased." That's a weaker form of Cessationism, but does offer some supplementary support.