What does the Bible say about tithing?
Before you answer please remember that tithing and giving are not the same thing.
(Skip to the last two paragraphs before the caveat for the final answer. The rest is the basis of the answer I gave.)
Tithing is mentioned only three or four times in the New Testament.
We know that salvation is by grace, through faith, and not of good works (Eph. 2:8), so tithing (or not) can neither add to nor take away from your salvation.
Tithing is little different than circumcision, which was also something demanded in Mosaic law. Acts 15 clearly puts to death the idea that we cannot be saved without following Mosaic law by using the issue of circumcision.
After considering the matter, they came to this conclusion:
Further, Jesus frees Israel from the law, and a new law is written on our hearts. (Hebrews 8:8-10)
This coincides beautifully with Hosea 6:6, which tells us, even in the Old Testament that God is more concerned with us loving Him and the condition of our hearts than in what we give.
Hosea 6:6: For I desired mercy, and not sacrifice; and the knowledge of God more than burnt offerings
And finally, pulling it all together, we have 2 Corinthians 9:7, where giving is addressed in the New testament:
Based on all of this, no we are not required to tithe under the New Testament, any more than we are required to be circumcised, sacrifice lambs, or the Mosaic law.
Don't forget what Jesus told us in Matthew 22:
In the New Testament, we are under the law of Love, not the law of Moses. The Law of love fulfills all of the requirements of the Mosaic law. (If we love God, we will have no other gods before Him. If we love our neighbor, we will not steal, bear false witness, covet, etc. If we love our parents, we will honor them, and so on.)
Giving of a cheerful heart, freely and because you want to is the replacement of the original tithe, and fulfills the requirement of the tithe, just as the law of Love fulfills the original Mosaic law. Not to mention that freely giving is a demonstration of love toward God, and a fulfillment of the New Testament law of Love.
Edit - added - Now for the caveat
We are required, in the new Testament to obey and submit to authority. (Hebrews 13:17, , Romans 13:1-7, Titus 3:1, and many others)
Your Church may require tithing, at which point, New Testament law WOULD require tithing as obedience to the authority. If this is the case, you're in a position where you need to make a decision.
I cringe to even state the last once, because these days, people will leave a Church any time they hear something they don't like preached. (2 Timothy 4:3) IMHO, you should only leave a Church if doctrine is unsound, and you know this Biblically.