The apostolic office ceased with the death of the last apostle in the first century - by Eric Davis

Found here. My comments in bold.
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While this article is about the LDS "prophet," it also contains an explanation as to why in the author's view there are no apostles. Perhaps he will provide us with a biblical case we seek for the ceasing of the apostolic.
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Scripture teaches that there are at least three qualifications necessary to be an apostle. First, an apostle had to be an eyewitness of the resurrected Christ (Acts 1:22; 10:39-41; 1 Cor. 9:1, 15:7-8).

(Since cessationists are routinely reluctant to quote their scripture references, our first task is to actually quote them.

Ac. 1:21 Therefore it is necessary to choose one of the men who have been with us the whole time the Lord Jesus went in and out among us, 22 beginning from John’s baptism to the time when Jesus was taken up from us. For one of these must become a witness with us of his resurrection.

Ac. 10:39 We are witnesses of everything he did in the country of the Jews and in Jerusalem.

1Co. 9:1 Am I not free? Am I not an apostle? Have I not seen Jesus our Lord? Are you not the result of my work in the Lord?

1Co. 15:7 Then he appeared to James, then to all the apostles, 8 and last of all he appeared to me also, as to one abnormally born.

The key verse is Ac. 1:22, which is the criteria set forth by the apostles as they sought to replace Judas Iscariot.  There is but a single criteria here: "been with us the whole time." The second statement, "become a witness with us of his resurrection," is the one to which the author appeals.  

The word "become" is relevant here.  The word is Ginomai, which means 

1. to become, i.e. to come into existence, begin to be, receive being
2. to become, i.e. to come to pass, happen
3. of events
4. to arise, appear in history, come upon the stage
5. of men appearing in public
6. to be made, finished
7. of miracles, to be performed, wrought
8. to become, be made

One doesn't BECOME a witness to, say, a crime. One IS a witness. Luke is not talking about the candidates seeing something. This isn't a case of the need to have witnessed the event of the resurrection or beheld the risen Christ. The author has read it too casually and is building a doctrine upon it. 

Luke is telling us that the new apostle needs to join with the rest of the apostles and embrace the ministry of re-telling of the resurrection! This is a duty of the new apostle, not a qualifier to become one. 

Only a few verses before, we read this: Ac. 1:8 But you will receive power when the Holy Spirit comes on you; and you will be my witnesses in Jerusalem, and in all Judea and Samaria, and to the ends of the earth. This is how to understand "witness."

In this light, the other verses cited become irrelevant.

Further, the first statement ["been with us the whole time"] disqualifies Paul. But since we know Paul was indeed an apostle, we must conclude that these criteria were not set forth as a template, but rather as what they considered important at the time. In fact, they might have been rushing to meet a perceived need. 

We do not read that the Holy Spirit spoke to them about this. Indeed, God was going to provide Paul as an apostle, as well as other apostles, in due time. It could very well be the eleven were acting with presumption.) 

Second, an apostle had to be directly appointed by Jesus Christ (Mark 3:14; Luke 6:13; Acts 1:2, 24; 10:41; Gal. 1:1).

Once again we quote the supplied scripture references:

Mk. 3:14 He appointed twelve — designating them apostles...

Lk. 6:13 When morning came, he called his disciples to him and chose twelve of them, whom he also designated apostles...

Ac. 1:2 until the day he was taken up to heaven, after giving instructions through the Holy Spirit to the apostles he had chosen.

Ac. 1:24 Then they prayed, “Lord, you know everyone’s heart. Show us which of these two you have chosen..."

Ac. 10:41 He was not seen by all the people, but by witnesses whom God had already chosen — by us who ate and drank with him after he rose from the dead.

Ga. 1:1 Paul, an apostle — sent not from men nor by man, but by Jesus Christ and God the Father...

As you read through these scriptures, do you see anywhere where apostles "had to be directly appointed?" The original twelve were of course, but where do we find any information that it had to be this way? Matthias wasn't. Neither was Barnabas. Same with Andronicus and Junias.)

Finally, an apostle had to confirm his message and ministry with miraculous apostolic gifts (Matt. 10:1-2; Acts 1:5-8; 2:43; 4:33; 5:12; 8:14; 2 Cor. 12:12; Heb. 2:3-4).

Again the actual quotes: 

Mt. 10:1 He called his twelve disciples to him and gave them authority to drive out evil spirits and to heal every disease and sickness.

Ac. 1:5-8 For John baptized with water, but in a few days you will be baptized with the Holy Spirit.” 6 So when they met together, they asked him, “Lord, are you at this time going to restore the kingdom to Israel?” 7 He said to them: “It is not for you to know the times or dates the Father has set by his own authority. 8 But you will receive power when the Holy Spirit comes on you; and you will be my witnesses in Jerusalem, and in all Judea and Samaria, and to the ends of the earth.”

Ac. 2:43 Everyone was filled with awe, and many wonders and miraculous signs were done by the apostles.

Ac. 4:33 With great power the apostles continued to testify to the resurrection of the Lord Jesus, and much grace was upon them all.

Ac. 5:12 The apostles performed many miraculous signs and wonders among the people.

Ac. 8:13 Simon himself believed and was baptized. And he followed Philip everywhere, astonished by the great signs and miracles he saw.

2Co. 12:12 The things that mark an apostle — signs, wonders and miracles — were done among you with great perseverance.

He. 2:3-4 how shall we escape if we ignore such a great salvation? This salvation, which was first announced by the Lord, was confirmed to us by those who heard him. 4 God also testified to it by signs, wonders and various miracles, and gifts of the Holy Spirit distributed according to his will.

Again, as you read through these scriptures, do you see anywhere where they say that only the apostles could do miracles? In fact, Stephen was not an apostle, but we read of him: Ac. 6:8 Now Stephen, a man full of God’s grace and power, did great wonders and miraculous signs among the people.

Regarding Mt. 10:1, perhaps the author might tell us about the 72, also sent out with power? They came back rejoicing that demons submitted to them.

Regarding Ac. 1:5-8, we find it particularly interesting that the author seems to be suggesting that only the apostles were baptized with the Holy Spirit and fire. Peter quoted John the Baptist here. John the Baptist was publicly baptizing, and spoke this to the large crowd assembled. We don't even know if any apostles-to-be were there. They weren't apostles when John the Baptist said this, which means that he was intending his remarks for all who could hear.

Further, did the apostles go to the ends of the earth? No, that was the mission of the Church. We have all received the baptism of the Holy Spirit, we all received power, we are all called to preach the gospel. This cannot be referring to the apostles only.

Regarding He. 2:3-4, notice that there are two categories: 1) signs, wonders and various miracles, and 2) gifts of the Holy Spirit distributed according to his will. Does this passage say that only the apostles received gifts? Does it say that only the apostles do signs, wonders, and various miracles? No and no.

The author is simply feeding his preconceptions.)

No one beyond the first century is able to meet these qualifications. (Preposterous, of course.)

Additionally, the apostles were used to complete the canon of Scripture. (Not all Scripture was written by apostles. The author of Hebrews is uncertain.  James, Mark, Luke, and Jude were not apostles.)

Endowed as Christ’s messengers, they were uniquely tasked to speak with the authority of Christ. Much of the apostolic instruction then became the 27 books of the New Testament. Therefore, if there are apostles today, then the canon needs to be reopened and continued. (No verse is cited for this facile assertion.)

But, there are not. With the cessation of the apostles came the cessation of the canon.

Finally, early church leaders did not view themselves as apostles, notwithstanding their proximity to the first century. Individuals such as Clement, Ignatius, Polycarp, and Justin Martyr saw themselves as disciples of Christ and church leaders, but not apostles. (We are looking for the biblical case, not what men thought.)

Wayne Grudem writes, “It is noteworthy that no major leader in the history of the church – not Athanasius or Augustine, not Luther or Calvin, not Wesley or Whitefield – has taken to himself the title of ‘apostle’ or let himself be called an apostle” (Systematic Theology, p. 911). Therefore, a system may claim the term “apostle” to its leadership, but it could not be said that they were apostles in the biblical sense, from God.The purpose of the apostles was to serve as the foundation-laying process of the New Testament Church in the first century (Eph. 2:20). Once the foundation was laid, no additional foundation was needed. (One purpose was to lay the foundation. Actually, to be the foundation. But that begs the question: What else did apostles do, and should we have people in the church doing them? 

We read in 1Co. 12:28:

And in the church God has appointed first of all apostles, second prophets, third teachers, then workers of miracles, also those having gifts of healing, those able to help others, those with gifts of administration, and those speaking in different kinds of tongues. 29 Are all apostles? Are all prophets? Are all teachers? Do all work miracles? 30 Do all have gifts of healing? Do all speak in tongues? Do all interpret? 31 But eagerly desire the greater gifts. 

Notice that Paul supplies a list of things God has appointed in the church. He then concludes that we should desire the greater gifts. Apostles are included in the list of gifts we should desire. The author errs in considering all apostles to be occupying an office [as suggested in the title of the article] when in the NT, God is giving gifts to the church. And the gifts did not cease.)

Consequently, there have been no apostles since the death of the last apostle (likely John) in the first century. The church is now overseen by pastors (1 Tim. 3:1-7, Titus 1:5-9, 1 Pet. 5:1-3). (Oh, my. That's quite a howler. Yet again we quote the scriptures:

1Ti. 3:1 Here is a trustworthy saying: If anyone sets his heart on being an overseer,  he desires a noble task.

Tit. 1:6-7 An elder must be blameless, the husband of but one wife, a man whose children believe and are not open to the charge of being wild and disobedient. 7 Since an overseer is entrusted with God’s work, he must be blameless...

1Pe. 5:1-2 To the elders among you, I appeal as a fellow-elder, a witness of Christ’s sufferings and one who also will share in the glory to be revealed: 2 Be shepherds of God’s flock that is under your care, serving as overseers...

It only takes a casual reading of these scriptures to discover that pastors are not mentioned, nor are they being discussed. And overseer is an elder is a shepherd.  1 Peter 5:1-2 in particular says that the church is to be led by a plurality of elders. 

It's this kind of sloppy thinking that has led to errors and heresies of all kinds. We need to abandon our templates and false assumptions and read the Bible with the insight of the Holy Spirit informing us.)