Was Paul advocating killing a man because of sexual immorality? 1 Corinthians 5:5 (NIV)


  5 hand this man over to Satan for the destruction of the flesh, so that his spirit may be saved on the day of the Lord.


Here we have Paul referring to a man that was sleeping with 'his father's wife'. It seems from the way the phrase reads that Paul would see this man's flesh destroyed, if not directly by the members of the Corinthian church then indirectly, through the agency of Satan.

What is the meaning of the phrase 'destruction of the flesh'? If Paul is advocating killing him then what are the ramifications?
No.  Paul was advocating kicking him out of the Church, not killing him.  

The fact that it means to kick the person out of the Church is clarified in verse 12: (Emphasis added.)


  12 What business is it of mine to judge those outside the church? Are
  you not to judge those inside? 13 God will judge those outside. “Expel
  the wicked person from among you.”


The way I understand it is this: The idea is to not turn a blind eye to sin, but to confront it and deal with it, particularly within the body of the Church.  To turn a blind eye to sin does nothing but encourage it.  To confront it and make it clear that it is not to be tolerated within the Church is better, because it forces the sinner to either acknowledge the sin and repent, or leave the Church, where they will not pollute it further. 

A person cannot repent if they are put to death, so this clearly cannot be stating that we are to put the sinner to death.

There is a nice version of this verse here that clarifies the meaning of the original Greek word that is translated as "the flesh".  Due to language difference and cultural differences, the original meaning often gets lost in translation.

http://www.biblegateway.com/passage/?search=1+Corinthians+5&amp;version=NIV

Footnote below: 


  1 Corinthians 5:5 In contexts like this, the Greek word for flesh
  (sarx) refers to the sinful state of human beings, often presented as
  a power in opposition to the Spirit.