What does the Bible say about tithing? Do we have to tithe? Why does God need us to tithe? What are the results of NOT tithing?
The strongest words with respect to the benefits of paying tithing, and the consequences for not doing so, can be found in Malachi 3: 8-12:


  8 Will a man rob God? Yet ye have robbed me. But ye say, Wherein have
  we robbed thee? In tithes and offerings.
  
  9 Ye are cursed with a curse: for ye have robbed me, even this whole
  nation.
  
  10 Bring ye all the tithes into the storehouse, that there may be
  meat in mine house, and prove me now herewith, saith the Lord of
  hosts, if I will not open you the windows of heaven, and pour you out
  a blessing, that there shall not be room enough to receive it.
  
  11 And I will rebuke the devourer for your sakes, and he shall not
  destroy the fruits of your ground; neither shall your vine cast her
  fruit before the time in the field, saith the Lord of hosts.
  
  12 And all nations shall call you blessed: for ye shall be a
  delightsome land, saith the Lord of hosts.


However, it appears that in the Christian period, just as with many other provisions of the Law, tithing was superseded by a higher law.  Towards the end of both Acts 2 and Acts 4 it explains that the Christian community had all things in common, consecrating not a tithe (tenth) of their belongings but instead everything they owned to the general welfare of the church in order to do away with poverty.  (It's worth noting that, while there are many obvious differences and incompatibilities between Christianity and communism, Karl Marx's famous "from each according to his ability, to each according to his need" was inspired by this part of Acts.)

The location of the chapter break between Acts 4 and Acts 5 is unfortunate, as Acts 5 begins with a direct continuation of the discussion of the treatment of the consecration of property that Acts 4 ended with.  Here we see the story of a man and his wife who sold their property and gave part of the money to the Apostles, but kept back part of it to enrich themselves.  Apparently the Lord takes this very seriously, (Galatians 6: 7--God is not mocked,) as He struck both of them dead when they tried to reaffirm their lie when questioned by Peter!

I'm not aware of any church today that practices this doctrine of total consecration, but the principle is still there.  At the very least, a true believer should be willing to give up all that he posesses if it is needed to further the Lord's cause.  (See Matthew 13: 44-46, Luke 18: 18-25, and especially Luke 18: 29-30 for the Savior's own words on this matter.  See also Luke 12: 34/Matthew 6: 21--"Where your treasure is, there will your heart be also."  The entire point of the Gospel is to improve our hearts and turn them towards heaven instead of the things of this world.)