According to the New Testament, believers should give to the church. However, the specificity of tithing is not an area of emphasis in that part of the Bible, as in the Old Testament. As with many situations in the Old Testament, the New Testament offers a less legalistic perspective.
Matthew 23:23 actually makes fun of the Jewish religious leadership for its close adherence to the rules, including the tithe. Luke 18:12, part of one of Jesus' parables, critiques the self-righteous Pharisee's sense of superiority resulting from the fact that he fasts and tithes.
However, this should not be taken to mean that church members should not give. Instead, New Testament believers should not adhere rigidly to a set rule. They should give as they are able and feel led to by the Holy Spirit. Hebrews 7 details the story of Melchizedek, a priest to whom the Old Testament patriarch Abraham paid an offering. The purpose of that chapter is to demonstrate that Christ's priesthood is superior to that of the Levites and to teach believers that Christ's ministry is considerably superior to that of the Levites.
The spirit of New Testament giving appears in such passages as Acts 11:27-39, in which the disciples each gave according to their means, rather than focusing on a set percentage. Everyone gave as much as he could without thinking that 10 percent was a minimum or a maximum.