Well, it all depends on how one defines "Reformed." Some people limit it to the system of doctrine taught in the various paedobaptist creedal pronounces from the 16th and 17th Centuries (e.g., the Westminster Confession, Three Forms of Unity, Second Helvetic Confession, etc.). In other words, the Reformed
must hold to a covenant theology that demands paedobaptism.
Others define "Reformed" more broadly to accept Baptists. This would be best represented in the London Baptist Confession of Faith (1689). In this instance, Reformed theology is limited more to a soteriological Calvinism and a varied understanding of Covenant theology.
Still others hold the "Reformed" only refers to the so-called "five points of Calvinism." In this case, even Dispensationalists such as John MacArthur are "Reformed."
Within each of these camps there are indeed Reformed theologians who are charismatics. Most of them, however, are not in the first camp (the paedobaptists). Take, for examples, John Piper, Wayne Grudem, C.J. Mahaney, and others. All of these guys affirm the present operation of the charismatic gifts of tongues, interpretation of tongues, and prophecy.
So, if one defines "Reformed" more broadly, then no, Reformed theology does
not demand the cessation of the charismatic gifts. As for the first camp (the paedobaptists), it all depends on how you take the very first paragraph of the
Westminster Confession of Faith (paragraph I, section i). Notice the bolded statement:
Although the light of nature, and the works of creation and providence do so far manifest the goodness, wisdom, and power of God, as to leave men inexcusable; yet are they not sufficient to give that knowledge of God, and of His will, which is necessary unto salvation: therefore it pleased the Lord, at sundry times, and in divers manners, to reveal Himself, and to declare that His will unto His Church; and afterwards, for the better preserving and propagating of the truth, and for the more sure establishment and comfort of the Church against the corruption of the flesh, and the malice of Satan and of the world, to commit the same wholly unto writing; which maketh the Holy Scripture to be most necessary; those former ways of God's revealing His will unto His people being now ceased.
To what exactly does this refer, and does it include tongues and prophecy as exhibited in the New Testament? I don't know, to be honest. This statement seems at its core to be denying the possibility of further revelation
a la inspired Scripture. Whether one can take it further will be a matter of debate. The Westminster divines did, outside of the Anabaptists, to my knowledge did not have anything comparable to the Pentecostal movement as we see it presently in the United States and elsewhere.
Here is an interesting statement by John Calvin, just for kicks:
Those who preside over the government of the church in accordance with Christ’s institution are called by Paul as follows: first apostles, then prophets, thirdly evangelists, fourthly pastors, and finally teachers (Ephesians 4:11). Of these only the last two have an ordinary office in the church; the Lord raised up the first three at the beginning of his Kingdom, and now and again revives them as the need of the times demands.
—John Calvin, Institutes of the Christian Religion, IV.iii.4.
So, it seems that Calvin takes a reasonable position over against staunch cessationism that we see today. However, he also seems elsewhere to clarify his position:
...in the first letter to the Corinthians (1 Corinthians 12:28), he lists others, as powers, the gift of healing, interpretation, government, and caring for the poor. Two of these I omit as being temporary, for it is not worthwhile to tarry over them. But two of them are permanent: government and caring for the poor."
—John Calvin, Institutes of the Christian Religion, IV.iii.8.
Hopefully this helps. My conclusion for the time being would be that, no, being Reformed does not
demand that one also be a cessationist, although that is by far the most common viewpoint among the Reformed.