What is the Biblical basis for moral absolutism? In this world of moral relativism, do any Christians make a scriptural (Biblical) case for moral absolutism? That is, how do they use the Bible to make the case that there is such a concept as absolute "right" and "wrong"?
The existence of Moral absolutism is actually fairly easily identified biblically in verses such as:


  Then the LORD God said, "Behold, the man has become like one of us in knowing good and evil. Now, lest he reach out his hand and take also of the tree of life and eat, and live forever-" - Genesis 3:22 ESV


In which God specifies that 


Good and evil exist, as separate and opposite attributes.  
They can both be known (not evaluated or judged), and therefore are not relative. 


Also communicated without relativism:


  See, I have set before you today life and good, death and evil. - Deuteronomy 30:15 ESV


And note here that Solomon asks for discernment, recognizing that he is not the arbiter of that decision (which pleases the Lord):


  Give your servant therefore an understanding mind to govern your people, that I may discern between good and evil, for who is able to govern this your great people?" - 1 Kings 3:9 ESV


What is sometimes less easily identifiable is whether or not actions that carry an absolute moral value are treated in the same manner.  Often they are not due to the fact that God is forgiving.  David writes:


  1 Blessed is the one whose transgression is forgiven, whose sin is covered. 
  2 Blessed is the man against whom the LORD counts no iniquity, and in whose spirit there is no deceit.  - Psalm 32:1-2 ESV 


Sin is always sin, and evil is always evil, but God has provided a means for the faithful to seek forgiveness.  

Before Christ (as one example):


  27 "If anyone of the common people sins unintentionally in doing any one of the things that by the LORD's commandments ought not to be done, and realizes his guilt,
  28 or the sin which he has committed is made known to him, he shall bring for his offering a goat, a female without blemish, for his sin which he has committed.
  29 And he shall lay his hand on the head of the sin offering and kill the sin offering in the place of burnt offering.
  30 And the priest shall take some of its blood with his finger and put it on the horns of the altar of burnt offering and pour out all the rest of its blood at the base of the altar.
  31 And all its fat he shall remove, as the fat is removed from the peace offerings, and the priest shall burn it on the altar for a pleasing aroma to the LORD. And the priest shall make atonement for him, and he shall be forgiven. - Leviticus 4:27-31 ESV 


After Christ's resurrection, and through faith in Him:


  If we confess our sins, he is faithful and just to forgive us our sins and to cleanse us from all unrighteousness. - 1 John 1:9


Also contributing to confusion is the timetable by which God chooses to punish those who have sinned against him.  Some are stuck down immediately (Uzzah is a good example), while others will live a full life of sin before facing judgement.  Again, the determination of good / evil is not realitive even though the punishment can be.  

Finally, the biggest contributor masking the absolute moralism present in scripture is the inherent difference between true good / true evil and law.  Good / evil morality is unchanging, however laws (including God's laws for man) can be created, edited, or dissolved as needed to fit a particular people at a particular time.  God forbidding the people of Israel from wearing garments of different cloths is not inherently immoral, for example.  This subject of Covenant theology or Dispensational theology addressing the transformation of God's law for man is a discussion beyond the scope of this question, but for the sake of sticking with biblical references to the argument, Jesus establishes a new law for Man in his time on Earth:


  A new commandment I give to you, that you love one another: just as I have loved you, you also are to love one another. - John 13:34


To summarize, God tells us biblically that good and evil exist without variability, but his dealings with us in forgiveness, punishment, and law are according to his will.  Because his will is not ultimately knowable, how are we to navigate this path?  By seeking his guidance in reading scripture and prayer:


  5 Trust in the LORD with all your heart, and do not lean on your own understanding. 
  6 In all your ways acknowledge him, and he will make straight your paths. 
  7 Be not wise in your own eyes; fear the LORD, and turn away from evil.  - Proverbs 3:5-7 ESV


And 


  But from there you will seek the LORD your God and you will find him, if you search after him with all your heart and with all your soul. - Deuteronomy 4:29 ESV


And


  7 "Ask, and it will be given to you; seek, and you will find; knock, and it will be opened to you.
  8 For everyone who asks receives, and the one who seeks finds, and to the one who knocks it will be opened. - Matthew 7:7-8 ESV


So according to the bible, and addressing the question asked, God, who is ultimately knowledgeable of what is absolutely good and evil, establishes law, punishment, and forgiveness as needed by man, and provides for us a means of seeking His will, knowledge and guidance of this absolute.