What is the Biblical basis for saying that we won't have a second chance to be saved after we die? A commonly held belief is that people need to accept Jesus as Lord during their lifetime on Earth and failing to do so will result in the individual's damnation. Is there any support for this "last call" doctrine? 

Since there is life after death, I have wondered if people will have an opportunity to make a decision for Jesus after their death. Is there any support for this "second chance" idea? I have tried to study it and have not able to find anything in the bible that would contradict this idea that we can still make a decision for Christ after this life.
The mainstream Christian answer is that there is no chance for repentance after death.  The Scripture used to support this is Hebrews 9:27 which says


  And just as it is appointed for man to die once, and after that
  comes judgment


The protestant answer goes further saying this:
(From http://www.gotquestions.org/second-chance-salvation.html)


  While the idea of a second chance for salvation is appealing, the
  Bible is clear that death is the end of all chances. Hebrews 9:27
  tells us that we die, and then face judgment. So, as long as a person
  is alive, he has a second, third, fourth, fifth, etc. chance to accept
  Christ and be saved (John 3:16; Romans 10:9-10; Acts 16:31). Once a
  person dies, there are no more chances. The idea of purgatory, a place
  where people go after death to pay for their sins, has no biblical
  basis, but is rather a tradition of the Roman Catholic Church


Now in all fairness, the Cathollic Church doesn't teach that there is a chance for repentance after death.  This is a misrepresentation.

From the Catholic Answers forum:


  The teaching of the Church is very clear on this point: if we die in a
  State of Grace, we will attain salvation. Otherwise, we (probably) go
  to hell.
  
  We are placed in a State of Grace through Christian Baptism. We remove
  ourselves through mortal sin. We are restored to a State of Grace by
  Sacramental Confession.
  
  The Church recognizes the possibility that salvation might be attained
  by other means (which have not been revealed). Thus, we have hope for
  salvation for unbaptized infants (but nobody knows for sure what
  happens to them)
  
  There's no telling what we can make of near-death experiences. If we
  accept them at face value, they could be interpreted as an experience
  of purgatory (not hell). Or they might represent some "other means" of
  salvation that has not been revealed to the Church.


LDS teaching is a bit more... Complex.  Some LDS Scriptures seem to say Yes and some say no.

LDS Scripture teaching no repentance after death:


  Mosiah 2:36-39 
  
  And now, I say unto you, my brethren, that after ye have known and have been taught all these things, if ye should transgress and go
  contrary to that which has been spoken, that ye do withdraw yourselves
  from the Spirit of the Lord, that it may have no place in you to guide
  you in wisdom's paths that ye may be blessed, prospered, and
  preserved—I say unto you, that the man that doeth this, the same
  cometh out in open rebellion against God; therefore he listeth to obey
  the evil spirit, and becometh an enemy to all righteousness;
  therefore, the Lord has no place in him, for he dwelleth not in unholy
  temples. Therefore if that man repenteth not, and remaineth and dieth
  an enemy to God, the demands of divine justice do awaken his immortal
  soul to a lively sense of his own guilt, which doth cause him to
  shrink from the presence of the Lord, and doth fill his breast with
  guilt, and pain, and anguish, which is like an unquenchable fire,
  whose flame ascendeth up forever and ever. 
  
  Alma 34:32-35 
  
  32 For behold, this life is the time for men to prepare to meet God; yea, behold the day of this life is the day for men to perform
  their labors. 33 And now, as I said unto you before, as ye have had so
  many witnesses, therefore, I beseech of you that ye do not
  procrastinate the day of your repentance until the end; for after this
  day of life, which is given us to prepare for eternity, behold, if we
  do not improve our time while in this life, then cometh the night of
  darkness wherein there can be no labor performed. 34 Ye cannot say,
  when ye are brought to that awful crisis, that I will repent, that I
  will return to my God. Nay, ye cannot say this; for that same spirit
  which doth possess your bodies at the time that ye go out of this
  life, that same spirit will have power to possess your body in that
  eternal world. 35 For behold, if ye have procrastinated the day of
  your repentance even until death, behold, ye have become subjected to
  the spirit of the devil, and he doth seal you his; therefore, the
  Spirit of the Lord hath withdrawn from you, and hath no place in you,
  and the devil hath all power over you; and this is the final state of
  the wicked.


LDS teaching stating there can be repentance after death...


  Doctrine and Covenants 128:5
  
  5 You may think this order of things to be very particular; but let me
  tell you that it is only to answer the will of God, by conforming to
  the ordinance and preparation that the Lord ordained and prepared
  before the foundation of the world, for the salvation of the dead who
  should die without a knowledge of the gospel.


All that aside, getting back to a purely Biblical answer, we fall back to the standard Protestant "No. Once you die there is no further chance for repentance, according tothe teachings of Jesus".  For this, we need look no further than Luke 16:19-31


  19 “There was a rich man who was dressed in purple and fine linen and
  lived in luxury every day. 20 At his gate was laid a beggar named
  Lazarus, covered with sores 21 and longing to eat what fell from the
  rich man’s table. Even the dogs came and licked his sores.
  
  22 “The time came when the beggar died and the angels carried him to
  Abraham’s side. The rich man also died and was buried. 23 In Hades,
  where he was in torment, he looked up and saw Abraham far away, with
  Lazarus by his side. 24 So he called to him, ‘Father Abraham, have
  pity on me and send Lazarus to dip the tip of his finger in water and
  cool my tongue, because I am in agony in this fire.’
  
  25 “But Abraham replied, ‘Son, remember that in your lifetime you
  received your good things, while Lazarus received bad things, but now
  he is comforted here and you are in agony. 26 And besides all this,
  between us and you a great chasm has been set in place, so that those
  who want to go from here to you cannot, nor can anyone cross over from
  there to us.’
  
  27 “He answered, ‘Then I beg you, father, send Lazarus to my family,
  28 for I have five brothers. Let him warn them, so that they will not
  also come to this place of torment.’
  
  29 “Abraham replied, ‘They have Moses and the Prophets; let them
  listen to them.’
  
  30 “‘No, father Abraham,’ he said, ‘but if someone from the dead goes
  to them, they will repent.’
  
  31 “He said to him, ‘If they do not listen to Moses and the Prophets,
  they will not be convinced even if someone rises from the dead.’”


This horrifying account tells of someone who has died in his sin, and has no way out.  Verse 25 makes it pretty clear that there is no way across that chasm from Hell to Heaven after death.