Irenaeus seems to have written shortly after Hegesippus, and he comes much closer to Dave Armstrong's view of apostolic succession than any previous source. But that isn't saying much.
Irenaeus believed in a form of Roman primacy, but not a papacy. Even Roman Catholic scholars have acknowledged that the passage Catholics most often cite from Irenaeus on the subject (Against Heresies, 3:3:2) has been abused in support of Catholicism. For example:
"All churches must agree with it [the Roman church] on matters of doctrine because they must agree with the apostolic tradition preserved by the apostolic churches....In any event this is a striking testimony though not, in my view, as decisive as some have argued. The context of Irenaeus' argument does not claim that the Roman Church is literally unique, the one and only in its class; rather, he argues that the Roman church is the outstanding example of its class, the class in question being apostolic sees. While he chose to speak primarily of Rome for brevity's sake, in fact, before finishing, he also referred to Ephesus and Smyrna....The German Catholic scholar, Norbert Brox of Regensburg, has claimed that the argument is framed entirely within a western context. At first I found this argument weak, but after comparing Irenaeus' argument to its expansion as found in Tertullian's De praescriptione haereticorum (36), (cf. next chapter), I find Brox's argument more convincing." (Robert Eno, The Rise Of The Papacy [Wilmington, Delaware: Michael Glazier, 1990], pp. 39-40)
"It is indeed understandable how this passage has baffled scholars for centuries! Those who were wont to find in it a verification of the Roman primacy were able to interpret it in that fashion. However, there is so much ambiguity here that one has to be careful of over-reading the evidence....Karl Baus' interpretation [that Irenaeus wasn't referring to a papacy] seems to be the one that is more faithful to the text and does not presume to read into it a meaning which might not be there. Hence, it neither overstates nor understates Irenaeus' position. For him [Irenaeus], it is those churches of apostolic foundation that have the greater claim to authentic teaching and doctrine. Among those, Rome, with its two apostolic founders, certainly holds an important place. However, all of the apostolic churches enjoy what he terms 'preeminent authority' in doctrinal matters." (William La Due, The Chair Of Saint Peter [Maryknoll, New York: Orbis Books, 1999], p. 28)
The historian Eric Osborn, in a recent study of Irenaeus, concludes:
"The subjection of all churches to Rome would be unthinkable for Irenaeus." (Irenaeus Of Lyons [New York: Cambridge University Press, 2005], p. 130)
The Roman primacy Irenaeus refers to is a result of non-papal factors, such as the Roman church's historical relationship with two prominent apostles, its familiarity to other churches, and probably its location in the capital of the empire. Irenaeus believed in a form of Roman primacy that doesn't imply a papacy.
Why are Catholics going to this passage in Irenaeus to begin with? A few hundred pages of Irenaeus' writings are extant, and we have descriptions of some of his non-extant writings. He frequently addressed issues of authority, repeatedly appealing to the authority of the apostles, the authority of those who knew the apostles, the authority of scripture, etc. He never appeals to papal authority, nor does he ever even mention it. Yet, Catholics so often tell us that the papacy is the foundation of the church, the center of unity, that it's the solution to a wide variety of problems in Protestantism and Eastern Orthodoxy, etc. How likely is it that Irenaeus would have believed in the concept of a papacy, yet would have said so little of it? The fact that discussions of the papacy in Irenaeus place so much emphasis on this one passage, which doesn't actually say anything of a papacy, is revealing.
In the same section of his treatise, Irenaeus goes on to refer to the importance of the churches in Smyrna and Ephesus, and the reasons he gives for the prominence of those churches have nothing to do with a permanent status established by Jesus and the apostles. Other sources before and around the time of Irenaeus, such as Ignatius of Antioch and Tertullian, give non-papal reasons for the importance of the Roman church. Irenaeus probably held a high view of that church for similar reasons, and the same can be said of his high view of Smyrna and Ephesus.
Notice that the opening segment of this section in Irenaeus (Against Heresies, 3:3:1) gives a practical explanation for the significance of the apostolic churches and their bishops. He says nothing about Matthew 16, an office established by Jesus, infallibility, etc. Rather, Irenaeus is (correctly) appealing to these churches' (and their bishops') historical proximity to the apostles. His reasoning is much like what we see today in Christian apologetics, an appeal to concepts like the earliness of a source, geographical proximity to an event in dispute, and eyewitness testimony. Irenaeus is presenting us with a historical argument that any Protestant could accept. No papacy is involved.
He mentions bishops in all three churches (Rome, Smyrna, and Ephesus), although he doesn't name any of them in the case of Ephesus. His focus is on churches, not bishops. There's a difference between a non-jurisdictional primacy of the Roman church and a jurisdictional primacy of the Roman bishop. And there's a difference between a primacy that results from practical factors and a primacy that results from a permanent office established by Jesus and taught by the apostles. The evidence suggests that Irenaeus and other early sources had the former in view, even though Catholics read the latter into their comments.
In addition to Irenaeus' focus on the Roman church and its primacy for non-papal reasons, note that he repeatedly refers to Peter and Paul together, without placing Peter in a position of higher authority. (He mentions Peter before Paul here, but that sort of ordering is inconclusive, and he reverses the two, mentioning Paul first, elsewhere.) He repeatedly refers to how the Roman church reflects the traditions of the apostles (plural). The apostles Peter and Paul (plural) founded the Roman church. They (plural) appointed Linus as bishop of Rome. Clement is referred to as the Roman bishop appointed in third place from the apostles (plural), Sixtus is referred to as the sixth from the apostles (plural), and Eleutherius is referred to as the twelfth from the apostles (plural). Clement is commended for his knowledge of the traditions of the apostles (plural). The Roman church is commended for reflecting the traditions of the apostles (plural) in a document we today call First Clement. Irenaeus had many opportunities to assert a jurisdictional primacy of Peter. He never does it. He doesn't even refer to a non-jurisdictional primacy. Over and over, he places Peter and Paul together. There's no reason to conclude that he viewed Peter and the bishops of Rome as Popes. The foundational doctrine of Catholicism, the papacy, is unknown to Irenaeus.
In my next post in this series, I'll address some of Dave Armstrong's comments about Irenaeus and Roman primacy.