A great many of the statements by the Apostolic Fathers on the afterlife are verbatim quotes or close paraphrases of Biblical texts. While these are very helpful for an understanding of what texts were considered authoritative by early Christians, they are less helpful for purposes of this question–that is, providing additional commentary on how the Biblical text was understood.
Bipartite vs. Tripartite natures of man
I am unaware of any statement by the Apostolic Fathers that substantially enlarges upon what is described in the Biblical text.
Clement of Rome makes the following potentially relevant statement:
Finally may the All seeing God and Master of spirits and Lord of
all
flesh, who chose the Lord Jesus Christ, and us through Him for a peculiar people, grant unto every soul that is called after His
excellent and holy Name faith (1 Clement 64:1)
But whether soul = spirit + body or is a separate entity is not described by Clement.
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Soul sleep vs. Consciousness during the intermediate state
Perhaps the most direct statement on the matter is made in the Shepherd of Hermas:
the apostles and the teachers who preached the name of the Son of God,
after they had fallen asleep in the power and faith of the Son of God,
preached also to them that had fallen asleep before them (Hermas 92:5)
This passage contains clear, conscious echoes of 1 Peter 3:18-20, but expands upon it to teach that not only Jesus, but His followers, taught the dead in the intermediate state (the subsequent verses indicate this is pre-resurrection). This concept is well-supported by Ante-Nicene Fathers of later generations.
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Clement of Rome supports the contemporary Jewish & Christian view (such as is taught in the parable of the rich man and Lazarus–see Luke 16) that there is an intermediate state in which the righteous receive some form of glory/rest:
There was Peter who by reason of unrighteous jealousy endured not one not one but many labors, and thus having borne his testimony
went to his appointed place of glory.
By reason of jealousy and strife Paul by his example pointed out the prize of patient endurance. After that he had been seven times in
bonds, had been driven into exile, had been stoned, had preached in
the East and in the West, he won the noble renown which was the reward
of his faith,
having taught righteousness unto the whole world and having reached the farthest bounds of the West; and when he had borne his testimony
before the rulers, so he departed from the world and went unto the
holy place, having been found a notable pattern of patient
endurance. (1 Clement 5:4-6)
Here we have an apostolic father who is suggesting that although Peter & Paul have not yet been resurrected (see 1 Clement 24:1 & 26:1), they are already in a better holier place and have already received some form of glory. Clearly Clement believes there is something between death and the resurrection.
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Polycarp supports (and complicates!) post-mortal consciousness:
I exhort you all, therefore, to yield obedience to the word of
righteousness, and to exercise all patience, such as you have seen
[set] before your eyes, not only in the case of the blessed Ignatius,
and Zosimus, and Rufus, but also in others among yourselves, and in
Paul himself, and the rest of the apostles. [This do] in the assurance
that all these have not run in vain,
but in faith and righteousness, and that they are [now] in their due
place in the presence of the Lord (Epistle to the Philippians,
chapter 9)
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Annihilationism / Conditional immortality vs. Immortality of the Soul
No clear statement in support of annihilationism is to be found in the Apostolic Fathers.
Glenn Peoples has argued for annihilationism by implication in the writings of Ignatius of Antioch (see here), but as noted by Talbot, the first direct statement for annihilationism by a Christian comes from Arnobius of Sicca in the 4th century (see here)...and Arnobius’ views were considered heretical when he introduced them.
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Conclusion
The only topic raised in the OP where the Apostolic Fathers are abundantly clear is post-mortal consciousness--they are decidedly in favor of it.
Annihilationism does not find clear support in the Apostolic Fathers (but does find much opposition in the writings of their disciples), and proponents of bipartite & tripartite views are unlikely to find compelling evidence one way or the other. Whatever they conclude from reading the Bible will be relatively unchallenged by the writings of the Apostolic Fathers.
Numerous relevant statements were made by Justin, Irenaeus, Clement of Alexandria, Tertullian, and Origen–however, I will exclude them from this post per the Meta guidance. They are ante-Nicene Fathers, but not Apostolic Fathers. A compilation of many of their relevant statements is found here.
