Tertullian probably is the second most cited patristic source on apostolic succession, after Irenaeus. Many of the points I made about Irenaeus could also be made about Tertullian. I won't be getting into a lengthy comparison between Tertullian's beliefs and those of Roman Catholicism, as I did with Irenaeus. Those interested in some examples of how Tertullian's beliefs differed from those of Catholics can find some in my articles here. What I want to do in this post is give some examples of the qualifications Tertullian included in his concept of apostolic succession, qualifications that are problematic for the Catholic appeal to Tertullian on this subject.
Like Irenaeus, Tertullian appeals to a core set of doctrines, none of them unique to Roman Catholicism or contrary to Protestantism:
"Now, with regard to this rule of faith - that we may from this point acknowledge what it is which we defend - it is, you must know, that which prescribes the belief that there is one only God, and that He is none other than the Creator of the world, who produced all things out of nothing through His own Word, first of all sent forth; that this Word is called His Son, and, under the name of God, was seen 'in diverse manners' by the patriarchs, heard at all times in the prophets, at last brought down by the Spirit and Power of the Father into the Virgin Mary, was made flesh in her womb, and, being born of her, went forth as Jesus Christ; thenceforth He preached the new law and the new promise of the kingdom of heaven, worked miracles; having been crucified, He rose again the third day; then having ascended into the heavens, He sat at the right hand of the Father; sent instead of Himself the Power of the Holy Ghost to lead such as believe; will come with glory to take the saints to the enjoyment of everlasting life and of the heavenly promises, and to condemn the wicked to everlasting fire, after the resurrection of both these classes shall have happened, together with the restoration of their flesh. This rule, as it will be proved, was taught by Christ, and raises amongst ourselves no other questions than those which heresies introduce, and which make men heretics." (The Prescription Against Heretics, 13)
It's worth noting that Tertullian's rule of faith differs somewhat from that of Irenaeus. As the Roman Catholic patristic scholar Joseph Lienhard explains, "The rule of faith has no fixed form; each writer adapted it to his immediate goals and intent." (The Bible, The Church, And Authority [Collegeville, Minnesota: The Liturgical Press, 1995], p. 100) Not only did they differ in their definitions of the rule of faith, but they also differed in whether they appealed to a rule at all. Lienhard notes that Theodore of Mopsuestia "does not appeal to the rule of faith, as Irenaeus did" (p. 58). The Baptist patristic scholar D.H. Williams, writing about the later centuries of the patristic era, comments, "the notion of the Rule has become more broadly construed than in earlier centuries" (Tradition, Scripture, And Interpretation [Grand Rapids, Michigan: Baker Academic, 2006], p. 76). The historian Eric Osborn, commenting on the rule of faith in Irenaeus, writes, "The content of the rule of faith is entirely theological, without the ethical and ecclesiastical content, which it held in Paul and to which it returned in Augustine." (Irenaeus of Lyons [New York: Cambridge University Press, 2005], p. 149)
Tertullian is arguing for doctrines that were taught publicly and explicitly since the time of the apostles (The Prescription Against Heretics, 22, 25-27). He gives priority to what the apostles taught, making a point about the Marcionites' gospel similar to what Evangelicals have said about the Roman Catholic gospel:
"I am accustomed in my prescription against all heresies, to fix my compendious criterion (of truth) in the testimony of time; claiming priority therein as our rule, and alleging lateness to be the characteristic of every heresy....But should Marcion's gospel succeed in filling the whole world, it would not even in that case be entitled to the character of apostolic. For this quality, it will be evident, can only belong to that gospel which was the first to fill the world" (Against Marcion, 5:19)
Here are some more examples of Tertullian's reasoning, which is often reminiscent of what we find in Irenaeus:
"But what if a bishop, if a deacon, if a widow, if a virgin, if a doctor, if even a martyr, have fallen from the rule of faith, will heresies on that account appear to possess the truth? Do we prove the faith by the persons, or the persons by the faith?...We, however, are not permitted to cherish any object after our own will, nor yet to make choice of that which another has introduced of his private fancy. In the Lord's apostles we possess our authority; for even they did not of themselves choose to introduce anything, but faithfully delivered to the nations of mankind the doctrine which they had received from Christ. If, therefore, even 'an angel from heaven should preach any other gospel' than theirs, he would be called accursed by us....When, indeed, any man doubts about this, proof will be forthcoming, that we have in our possession that which was taught by Christ....But should they [the heretics] even effect the contrivance [of producing a list of bishops from the apostles], they will not advance a step. For their very doctrine, after comparison with that of the apostles, will declare, by its own diversity and contrariety, that it had for its author neither an apostle nor an apostolic man; because, as the apostles would never have taught things which were self-contradictory, so the apostolic men would not have inculcated teaching different from the apostles, unless they who received their instruction from the apostles went and preached in a contrary manner. To this test, therefore will they be submitted for proof by those churches, who, although they derive not their founder from apostles or apostolic men (as being of much later date, for they are in fact being founded daily), yet, since they agree in the same faith, they are accounted as not less apostolic because they are akin in doctrine....Since this is the case, in order that the truth may be adjudged to belong to us, 'as many as walk according to the rule,' which the church has handed down from the apostles, the apostles from Christ, and Christ from God, the reason of our position is clear, when it determines that heretics ought not to be allowed to challenge an appeal to the Scriptures, since we, without the Scriptures, prove that they have nothing to do with the Scriptures. For as they are heretics, they cannot be true Christians, because it is not from Christ that they get that which they pursue of their own mere choice, and from the pursuit incur and admit the name of heretics. Thus, not being Christians, they have acquired no right to the Christian Scriptures...I hold sure title-deeds from the original owners themselves, to whom the estate belonged. I am the heir of the apostles....Now, what is there in our Scriptures which is contrary to us?" (The Prescription Against Heretics, 3, 6, 9, 32, 37-38)
He's appealing to antiquity, consistency, and other common evidential concepts. A succession of bishops isn't enough without accompanying apostolic doctrine, and agreement with the apostolic faith is sufficient for those who don't have a succession of bishops.
Evangelicals agree with the apostolic rule of faith as it was often defined by patristic sources, like Tertullian. And Evangelicals are in fellowship with some of the modern groups that claim apostolic succession, like Anglicans. It's not as though we've rejected the core set of apostolic doctrines men like Irenaeus and Tertullian refer to, or have separated ourselves from all of those who claim a succession of bishops, just because we're not in fellowship with some such groups, like Catholicism and Orthodoxy.
Tertullian did believe that historical successions carried a lot of evidential significance, and he believed that the widespread acceptance of a doctrine was significant, for example. Those are old arguments that predate Tertullian and predate Christianity. Arguments from succession and popularity have been used by all sorts of groups in all sorts of contexts. Modern Roman Catholics do sometimes selectively appeal to similar arguments, but without the same qualifiers that Tertullian and other patristic sources added.