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.
Acts 15:5 But there rose up certain of the sect of the Pharisees which believed, saying, That it was needful to circumcise them, and to command them to keep the law of Moses.
6 And the apostles and elders came together for to consider of this matter.
After considering the matter, they came to this conclusion:
Acts 15: 19Wherefore my sentence is, that we trouble not them, which from among the Gentiles are turned to God:
20 But that we write unto them, that they abstain from pollutions of idols, and from fornication, and from things strangled, and from blood.
21 For Moses of old time hath in every city them that preach him, being read in the synagogues every sabbath day.
22 Then pleased it the apostles and elders with the whole church, to send chosen men of their own company to Antioch with Paul and Barnabas; namely, Judas surnamed Barsabas and Silas, chief men among the brethren:
23 And they wrote letters by them after this manner; The apostles and elders and brethren send greeting unto the brethren which are of the Gentiles in Antioch and Syria and Cilicia.
24 Forasmuch as we have heard, that certain which went out from us have troubled you with words, subverting your souls, saying, Ye must be circumcised, and keep the law: to whom we gave no such commandment:
Further, Jesus frees Israel from the law, and a new law is written on our hearts. (Hebrews 8:8-10)
8 For finding fault with them, he saith, Behold, the days come, saith the Lord, when I will make a new covenant with the house of Israel and with the house of Judah:
9 Not according to the covenant that I made with their fathers in the day when I took them by the hand to lead them out of the land of Egypt; because they continued not in my covenant, and I regarded them not, saith the Lord.
10 For this is the covenant that I will make with the house of Israel after those days, saith the Lord; I will put my laws into their mind, and write them in their hearts: and I will be to them a God, and they shall be to me a people:
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:
Every man according as he purposeth in his heart, so let him give; not grudgingly, or of necessity: for God loveth a cheerful giver.
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:
37 Jesus said unto him, Thou shalt love the Lord thy God with all thy heart, and with all thy soul, and with all thy mind.
38 This is the first and great commandment.
39 And the second is like unto it, Thou shalt love thy neighbour as thyself.
40 On these two commandments hang all the law and the prophets.
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.