As bradimus has indicated, this is a tricky issue, because projecting the modern debate of cessationism vs. continuationism onto church fathers is anachronistic. That said, some figures in the early church do talk about or infer a decline or end in at least some types supernatural occurrences, sometimes to what are often referred to as "spiritual gifts."
For this answer I'm leaning on two books for sources. The first, Charismatic Gifts in the Early Church, is written by a continuationist, Ronald A. N. Kydd. The second, Counterfeit Miracles, is by cessationist B. B. Warfield. They actually agree that spiritual gifts came to an end during the early church period, though they disagree on precisely when – Kydd suggests the middle of the third century, while Warfield argues for the first century.
We'll look at two categories of evidence – early church figures who indicate signs of a decline in supernatural gifts, followed by those who say that they have ended.
Kydd identifies two third-century figures who indicate that the use of spiritual gifts, once common and accepted, had begun to decline. The first is Cyprian, who relates objections to dreams and visions in his Epistle to Florentius Pupianus (no. 68, dated to about AD 254):
For I remember what has already been manifested to me, nay, what has been prescribed by the authority of our Lord and God to an obedient and fearing servant; and among other things which He condescended to show and to reveal, He also added this: “Whoso therefore does not believe Christ, who maketh the priest, shall hereafter begin to believe Him who avengeth the priest.” Although I know that to some men dreams seem ridiculous and visions foolish, yet assuredly it is to such as would rather believe in opposition to the priest, than believe the priest. (§10, emphasis added)
The second writer is more explicit. In several places in Against Celsus (AD 248), Origen refers to "traces" of workings of the Holy Spirit:
And there are still preserved among Christians traces of that Holy Spirit which appeared in the form of a dove. They expel evil spirits, and perform many cures, and foresee certain events, according to the will of the Logos. (1.46)
For they have no longer prophets nor miracles, traces of which to a considerable extent are still found among Christians, and some of them more remarkable than any that existed among the Jews; and these we ourselves have witnessed, if our testimony may be received. (2.8)
Moreover, the Holy Spirit gave signs of His presence at the beginning of Christ’s ministry, and after His ascension He gave still more; but since that time these signs have diminished, although there are still traces of His presence in a few who have had their souls purified by the Gospel, and their actions regulated by its influence. (7.8)
Thus we see that both of these authors believed that visible supernatural workings of the Spirit were still occurring in their day, but they indicate that a decline had taken place.
B. B. Warfield cites several authors who speak of an end to supernatural gifts, two of which are Augustine (354–430) and Chrysostom (349–407). In On the Profit of Believing, Augustine lists a variety of miracles done by Jesus, and says:
Why, say you, do not those things take place now? because they would not move, unless they were wonderful, and, if they were usual, they would not be wonderful. [...] They were done at a very suitable time, in order that, by these a multitude of believers having been gathered together and spread abroad, authority might be turned with effect upon habits. (§34)
Similarly, in On True Religion, 25.47:
When the Catholic Church had been founded and diffused throughout the whole world, on the one hand miracles were not allowed to continue till our time, lest the mind should always seek visible things, and the human race should grow cold by becoming accustomed to things which when they were novelties kindled its faith. (Augustine: Earlier Writings, Library of Christian Classics, 248)
In his Retractions, written significantly later in life, Augustine qualifies these statements, though he still says that some gifts have disappeared, specifically mentioning tongues and some healings:
For not even now, when a hand is laid on the baptized, do they receive the Holy Spirit in such a way that they speak with the tongues of all nations; nor are the sick now healed by the passing shadow of the preachers of Christ. Even though such things happened at that time, manifestly these ceased later. (1.12.7)
Warfield identifies numerous places that Chrysostom refers to the ceasing of miracles. For example:
Why, saith one, are there not now those who raise the dead, and perform cures? Yes, then, why, I say: why are there not now those who have a contempt for this present life? Do we serve God for hire? When man’s nature was weaker, when the Faith had to be planted, there were even many such; but now he would not have us to hang upon these signs, but to be ready for death. [...] For this cause it is that there are none such now; because that (future) life hath seemed to us honorless, seeing that for its sake we do nothing, whilst for this there is nothing we refuse to undergo. (Homily 8 on Colossians)
Even now there are some that seek them and say, Why do not miracles take place also at this present time? If you are faithful, as you ought to be, and lovest Christ as you ought to love Him, you have no need of signs, they are given to the unbelievers. (Homily 25 on John)
But, as mentioned in the beginning, we have to be careful not to conclude too much. Warfield writes:
Chrysostom fairly teems with expressions implying that miracle-working of every kind had ceased [...] and yet he records instances from his day! (47)
We see here several prominent examples of early church writers who indicate both a decline and end to visible workings of the Holy Spirit, such as prophecy, healings, and tongues. But we must be cautious so as not to conclude too much on this evidence alone. Warfield writes of the challenges involved:
It is a very disturbing fact further that the very Fathers who record long lists of miracles contemporary with themselves, yet betray a consciousness that miracles had nevertheless, in some sense or other, ceased with the Apostolic age. (46)
On the other hand, Ronald Kydd is comfortable marking an end of the period of charismatic gifts, based on this and other evidence. He writes:
[In the first half of the third century] it is clear that the importance granted to spiritual gifts was passing. [...] There came a point around AD 260 at which they no longer fitted in the highly organized, well-educated, wealthy, socially-powerful Christian communities. (87)