What is the biblical basis for NOT tithing? On what biblical grounds do denominations that do not adhere to the practice of tithing usually defend their case? What is the biblical basis for not tithing?
The biblical basis for Christians not being obliged to tithe is based on the fact that tithing is an Old Testament concept and applied only to people living under the Mosaic Law. The tithe was a requirement of the Law in which the Israelites were to give 10 percent of the crops they grew and the livestock they raised to the tabernacle/temple (Leviticus 27:30; Numbers 18:26; Deuteronomy 14:24; 2 Chronicles 31:5).
After the death and resurrection of Christ Jesus, the Law was fulfilled and God cancelled the written code, with its regulations (the Mosaic Law), and nailed it to the cross (Colossians 2:14).  The Mosaic Law required that tithes be collected from the people, but no longer:

The former regulation is set aside because it was weak and useless (for the law made nothing perfect), and a better hope is introduced, by which we draw near to God. (Hebrews 7:18)

Christians do not have to submit to any legalistic tithe system.  The New Testament nowhere designates a percentage of income a person should set aside, but the early church did collect money to send as a gift to Christians in Jerusalem:

On the first day of every week, each one of you should set aside a sum of money in keeping with his income, saving it up, so that when I come no collections will have to be made. (1 Corinthians 16:2)

Here is the biblical principle that applied to Christians back then and to Christians today:

Each man should give what he has decided in his heart to give, not reluctantly or under compulsion, For God loves a cheerful giver. (2 Corinthians 9:7)

Just as the first Christians sold their possessions and willingly shared what they had with those in need (Acts 4:34-35), so today Christians are at liberty to give freely, as they are able.