What is the Biblical basis against Apostolic Succession? Apostolic succession, in Christianity, the teaching that bishops represent a direct, uninterrupted line of continuity from the Apostles of Jesus Christ. According to this teaching, bishops possess certain special powers handed down to them from the Apostles; these consist primarily of the right to confirm church members, to ordain priests, to consecrate other bishops, and to rule over the clergy and church members in their diocese (an area made up of several congregations).(source).
Apostolic Succession means that the current ministers of the church derive their authority by unbroken succession from the apostles. Catholics, Orthodox and Anglicans accept this concept but it is denied by most Protestants.
What is the Biblical basis against Apostolic Succession?
Protestants are not generally against the idea of certain individuals having been called and gifted by God to exercise authority over a local assembly of believers, nor a hierarchy within that assembly, nor even a limited regional hierarchy within a denomination.  If this authority were what the idea of apostolic succession detailed there would be little difficulty.  It is the exercise of strong regional and ultimately global authority, which seems to be the ultimate outcome of apostolic succession as practiced by some traditions, that alarms most protestants.
Perhaps the biggest biblical basis against the passing of the apostolic office at a global level is the lack of biblical support for the notion.  There is clear and plenteous basis for the notion of guarding and passing on the apostolic message, which is the gospel of God, but the office of apostle is clearly a gifting of God, clearly held by multiple men at the same time and it is clearly not necessary for the succession to remain unbroken for the message to remain, for it is God who protects both his truth and his elect.
In both 1 Timothy and Titus we are given fairly specific lists of the attributes that Paul wants Timothy and Titus to look for in appointing overseers of churches.  Neither list indicates anything sounding like the passing of a mantle of apostolic authority.  In each list the qualifications appear as blameless character, good report, etc.  In fact, what is portrayed is the time tested fruit of the spirit in the lives of these men which qualify them as elders/bishops/overseers.

Therefore an overseer must be above reproach, the husband of one wife, sober-minded, self-controlled, respectable, hospitable, able to teach, not a drunkard, not violent but gentle, not quarrelsome, not a lover of money. He must manage his own household well, with all dignity keeping his children submissive, for if someone does not know how to manage his own household, how will he care for God's church? He must not be a recent convert, or he may become puffed up with conceit and fall into the condemnation of the devil. Moreover, he must be well thought of by outsiders, so that he may not fall into disgrace, into a snare of the devil. - 1 Timothy 3:2-7
This is why I left you in Crete, so that you might put what remained into order, and appoint elders in every town as I directed you— if anyone is above reproach, the husband of one wife, and his children are believers and not open to the charge of debauchery or insubordination. For an overseer, as God's steward, must be above reproach. He must not be arrogant or quick-tempered or a drunkard or violent or greedy for gain, but hospitable, a lover of good, self-controlled, upright, holy, and disciplined. He must hold firm to the trustworthy word as taught, so that he may be able to give instruction in sound doctrine and also to rebuke those who contradict it. - Titus 1:5-9

Just as Timothy was charged by Paul to exhort, rebuke, teach, and correct so Titus was commanded by Paul to do so with all authority.  Both of these men were to teach what they had seen and heard from Paul and that which was in accord with the Scriptures with the faith and love of Christ by the power of the indwelling Holy Spirit.

Follow the pattern of the sound words that you have heard from me, in the faith and love that are in Christ Jesus. By the Holy Spirit who dwells within us, guard the good deposit entrusted to you. - 2 Timothy 1:13-14

There is nothing in the Scripture that indicates some specialized process of seminary training (not that education is bad) that qualifies a person for leadership of a church.  Apostles, pastors, teachers, etc are called and gifted by God through the Holy Spirit which is given by and upon faith in the Lord Jesus Christ.  Men are then recognized within a particular body of believers for lives which demonstrate the truth they espouse and those men are set apart for a particular ministry.
The resistance to the kind of Apostolic Succession which leads to a consolidation of authority in one man was nothing newly begun in the Protestant Reformation.  Many of the Church Fathers wrote as early as the 3rd Century in opposition to the idea of the passing of the primacy of Peter; notably Tertullian, Origen, and Cyprian.  They considered an equality and unity of Bishops wherein each Bishop in the church in each city sat in 'Peter's chair&quot;:  Whomever sat in the 'chair' of the See in Rome held authority only over the Church in Rome rather than everywhere and whomever confessed Peter's confession of &quot;Thou art the Christ, the Son of the Living God&quot; was, by virtue of that confession, both 'rock' and 'key holder'.
There is one God, one faith, one baptism, and one spirit and if there is any connecting principle flowing from one Church leader to the next it is the very same principle which unites all those who are born again by the Grace of God through faith in the Lord Jesus Christ.