Did the Early Church (ante-Nicene period) believe in the continuation or cessation of the office of Prophet?

The early writers use the term "prophet" in differing ways.

Prophets are Old Testament Writers

Ignatius & Irenaeus appear most comfortable using "the prophets" in the Jewish sense of the phrase: "the prophets" refers to a portion of the Old Testament writers. They frequently refer to the prophets (OT) and the apostles (NT). On this use of the word, "the prophets" are seen as messengers of God's words in the past.

One of the classis examples is from Irenaeus, where he expresses in passing the idea that the prophets are no longer on the earth (he's talking about something else entirely, but this offhand comment says something about his worldview):

But the prophets also, when they were upon the earth... (Against Heresies 2.33.3)

2 Clement avoids this ambiguity by referring to "the books and the apostles" instead of "the prophets and the apostles", thereby not taking a position on the ongoing nature of the role of prophet (see 2 Clement 14:2).

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Support for ongoing existence of prophets

The Didache, on the other hand, supports some overlap between the roles of "apostle" and "prophet", and speaks of prophets as individuals that existed at the present time. In fact, it has an entire chapter (see Didache 11) dedicated to how one should treat a visiting prophet. My take on the Didache's presentation is that an apostle is an office in the church; a prophet is one who teaches prophetically. Ergo, a person could be both.

The Shepherd of Hermas expresses a belief in the ongoing existence of prophets, and speaks of a prophet as one who has "the divine spirit" or "the prophetic spirit", and "the man, being filled with the Holy Spirit, speaketh to the multitude, according as the Lord willeth" (see Mandate 11 1:7-9).

The martyrdom of Polycarp refers to Polycarp as "prophetic teacher" (see 16:2).

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Mixed reviews

Tertullian believed in ongoing prophetic revelation, though he was ostracized and ultimately considered a heretic by the Bishops of Rome (see discussion of Montanism here). So while this shows the Montanists believed in continuing prophetic revelation, it also shows that the majority of the Bishops in later years considered it false prophecy.

Irenaeus may have been comfortable using the term "the prophets" as a shorthand for the Old Testament authors, but he did acknowledge what could be termed "prophetic gifts":

For some do certainly and truly drive out devils...Others have foreknowledge of things to come: they see visions, and utter prophetic expressions. Others still, heal the sick by laying their hands upon them, and they are made whole. (Against Heresies 2.32.4)

John Wesley suggested a decline in such prophetic gifts and prophetic manifestations after the first few centuries of the Christian era:

It does not appear that these extraordinary gifts of the Holy Ghost were common in the Church for more than two or three centuries. We seldom hear of them after that fatal period when the Emperor Constantine called himself a Christian (see The Works of John Wesley Vol. 7 89:26)

Scholar Tad R. Callister summarized the views of a number of historians (and theologians like Wesley) in suggesting that many of the gifts or manifestations associated with prophets were still prevalent in the 2nd century, declined in frequency in the 3rd, and were notably more absent after that time (see The Inevitable Apostasy pp. 89-94 for a review of the relevant history).

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Conclusion

If "the prophets" refers to the writers of the Old Testament, the early church saw them as a feature of the past.

If "a prophet" is one who speaks God's words, the earliest writers (see especially the Didache & The Shepherd of Hermas, both written in either the late 1st or early 2nd century) see prophets as an ongoing part of God's work.

As the church centralized, there was increasing solidarity in rejecting sects & movements who were seen as adherents of false prophecy. This created a climate in which it was politically safer to stick with the dead prophets & apostles whose words were widely accepted (and would go on to become the Biblical text) than to espouse any contemporary individual as prophetic.