De corona militis
Tertullian's aim in this writing is to show that a Christian, if he is in the Roman army, may not wear the customary crown or wreath, as it amounts to idolatry.
"De corona militis" mentions prayers for the dead as an Apostolic ordinance ..." - Catholic Encyclopedia, Purgatory, subsection titled Tradition.The reference is apparently to a sentence in chapter 3, a chapter which is only a paragraph in its entirety. Because of its relative brevity, I will reproduce the entire paragraph to provide some context allowing the reader to gauge the accuracy of various comments on it:
 And how long shall we draw the saw to and fro through this line, when we have an ancient practice, which by anticipation has made for us the state, i.e., of the question? If no passage of Scripture has prescribed it, assuredly custom, which without doubt flowed from tradition, has confirmed it. For how can anything come into use, if it has not first been handed down? Even in pleading tradition, written authority, you say, must be demanded.  Let us inquire, therefore, whether tradition, unless it be written, should not be admitted. Certainly we shall say that it ought not to be admitted, if no cases of other practices which, without any written instrument, we maintain on the ground of tradition alone, and the countenance thereafter of custom, affords us any precedent. To deal with this matter briefly, I shall begin with baptism. When we are going to enter the water, but a little before, in the presence of the congregation and under the hand of the president, we solemnly profess that we disown the devil, and his pomp, and his angels.  Hereupon we are thrice immersed, making a somewhat ampler pledge than the Lord has appointed in the Gospel. Then when we are taken up (as new-born children), we taste first of all a mixture of milk and honey, and from that day we refrain from the daily bath for a whole week. We take also, in congregations before daybreak, and from the hand of none but the presidents, the sacrament of the Eucharist, which the Lord both commanded to be eaten at meal-times, and enjoined to be taken by all alike. As often as the anniversary comes round, we make offerings for the dead as birthday honours.  We count fasting or kneeling in worship on the Lord's day to be unlawful. We rejoice in the same privilege also from Easter to Whitsunday. We feel pained should any wine or bread, even though our own, be cast upon the ground. At every forward step and movement, at every going in and out, when we put on our clothes and shoes, when we bathe, when we sit at table, when we light the lamps, on couch, on seat, in all the ordinary actions of daily life, we trace upon the forehead the sign. (Tertullian, De Coronis Militis, Chapter 3 in entirety, emphasis added)
Tertullian indicated his purpose in his introductory notes in the previous chapter: to cite things held by custom or tradition that are not in Scripture. Reviewing the list, some of these ancient customs have survived into modern times, while others have not, such as refraining from bathing for a week after baptism or feeling pained when normal household bread falls upon the ground. At any rate, Tertullian mentions the practice of making offerings for the dead. This is not, however, mentioned as an Apostolic ordinance.
There is another place in the same writing which has some bearing on the question of whether the primitive church taught a doctrine of purgatory. Chapter 11 contains a discussion of whether it is permissible for a Christian to be a Roman soldier in the first place. He writes a chain of questions designed to show the incompatibility of being a Christian in the Roman army.
Shall it be held lawful to make an occupation of the sword, when the Lord proclaims that he who uses the sword shall perish by the sword? And shall the son of peace take part in the battle when it does not become him even to sue at law?He multiplies examples at some length, then comes to the one of interest for our present purposes:
And shall the Christian be burned according to camp rule, when he was not permitted to burn incense to an idol, when to him Christ remitted the punishment of fire?This quote, perhaps less advantageous to the idea that Tertullian spoke of purgatory, is not considered in the article.
De Monogamia, or On monogamy, is a writing that few Christians today would be comfortable endorsing. It was written while Tertullian was a Montanist, generally considered a heretical sect. The purpose of the writing is to show that it is morally wrong -- a breach of marriage vows and a rejection of monogamy -- for a widow to remarry. The teachings of Paul on remarriage are explicitly set aside in favor of a supposed new revelation of the Holy Spirit. For all of Tertullian's colorful views, he does still bear a kind of witness to the state of Christian teaching in the primitive church.
"... and in "De Monogamia" (cap. x, P. L., II, col. 912) he advises a widow "to pray for the soul of her husband, begging repose for him and participation in the first resurrection"; he commands her also "to make oblations for him on the anniversary of his demise," and charges her with infidelity if she neglect to succour his soul." - Catholic Encyclopedia, Purgatory, subsection titled Tradition.Here, then, is the applicable section of Tertullian's writing:
Indeed, she prays for his soul, and requests refreshment for him meanwhile, and fellowship (with him) in the first resurrection; and she offers (her sacrifice) on the anniversaries of his falling asleep. For, unless she does these deeds, she has in the true sense divorced him, so far as in her lies; and indeed the more iniquitously----inasmuch as (she did it) as far as was in her power----because she had no power (to do it); and with the more indignity, inasmuch as it is with more indignity if (her reason for doing it is) because he did not deserve it.I think few of us would accept Tertullian's characterization of any widow's actions after the burial of her husband as a "divorce"; the point of interest is that he this time mentions both prayers for the dead and offerings or sacrifices for the dead.
Did Tertullian speak of purgatory?
The question then remains: did Tertullian speak of purgatory? If purgatory is, as the Catholic Encyclopedia states, "a place or condition of temporal punishment for those who, departing this life in God's grace, are, not entirely free from venial faults, or have not fully paid the satisfaction due to their transgressions," then we must ask whether Tertullian has spoken of these things when we ask whether he spoke of purgatory. In the passages cited in the article on Purgatory, Tertullian does not mention a place or condition of punishment for the saved after death, or sinners paying the satisfaction due to their transgressions, or any reckoning of payment for sins apart from the cross of Christ or the final day of reckoning.
Among the interesting questions raised is the nature and purpose of the annual offerings for the departed. Is the offering simply the Mass, or is it something else? Was the intent to spare punishments in the present time as purgatory would suggest, or to have the person readier to participate in the resurrection, or to ease the mind and build the confidence of the living by the remembrance of Christ in the face of upcoming judgment? Similar questions arise about prayers for the dead: is the prayer intended as a plea for mercy in the face of coming judgment, or is the prayer intended as a plea for the mercy in the face of ongoing punishment? While Tertullian does not mention any of the key ingredients of purgatory, the standing practice of prayers for the dead came to play a role in the development of that doctrine.
In the final analysis, prayers for the dead are not incompatible with the doctrine of purgatory but not in themselves proof that the prayerful accept the basic premises of purgatory: that there is guilt not remitted for Christ's sake to those who inherit eternal life, and that such unremitted guilt works itself out as an ongoing punishment for the dead at some time between their deaths and the Last Day.