Here is a summary of the main arguments for and against Christian tithing (as I understand them). I conclude by affirming that Christians should tithe.
The Law of Moses has been fulfilled by Christ and believers are no longer under it (Rom 7:1-6; Eph 2:16; Col 2:14). We are rather told to rely on the Spirit and to fulfill the Law of Christ (1Cor 9:20-21; Gal 5:16-18), which is to love God and our neighbors—in so doing we fulfill the whole Law (Mt 22:37-37; Gal 5:13-15; 6:2). The New Testament authors certainly provide examples of what this will look like (E.g. Gal 5:17-23). Therefore, even though Christian morality consists in Spiritual living rather than law-keeping, our morality is not without objective definition. Nonetheless, though tithing is mentioned in the New Testament (e.g. Mt 23:23; Heb 7:5-9), it is nowhere clearly stated as an expectation for Christians.
It is true that Christians are not bound by the old Law of Moses. We are under the Law of Christ instead, keeping in step with the Spirit. (See references above.) However, it should be noted that tithing existed before the Law of Moses. Abraham tithed to Melchizedek (Gen 14:20; Heb 7:6). Jacob vowed to tithe to the Lord at Bethel (Gen 28:22). This practice was later codified in the Law. Therefore, tithing is a biblical practice that precedes the Law and exists on its own quite apart from the Law. True: the law requiring us to tithe has been nailed to the cross (Col 2:14). That does not mean, however, that we can escape the principle of tithing, which is a part of the heritage of all Abraham’s descendents.
A comparable example would be the Sabbath. The Sabbath was established at creation and set forth as a pattern for Adam and his progeny (Gen 2:2-3). It was later added to the Law of Moses and further elaborated (e.g. Ex 20:10; Lev 25). As Christians under the Law of Christ, we are not required to observe Sabbath days. In fact Paul loudly insists that we ignore anyone who tells us we are bound by this law (Col 2:16-17). Hebrews suggests that Christ has fulfilled the meaning of the Sabbath by achieving rest for us (Heb 4, cf. Mt 11:29-30). Nonetheless, we recognize that Sabbath is a principle quite apart from the Law and we are benefited by keeping it. Beginning in the New Testament era, Christians chose to keep one day of the week holy (Ac 20:7; 1Co 16:2). They chose Sunday, rather than Saturday, probably because the Lord rose on a Sunday (E.g. Mt 28:1; cf. Rev 1:10). In the same way, the legal requirement to tithe has been abolished but the principle precedes the law and it remains for the benefit of believers.
In a sense you could say, “Christians are under the law of tithing” if by “law” you meant a principle—as in the way you would say, “the law of gravity” or “the law of supply and demand.” Like these examples, the law of tithing is something we cannot escape. It is established by God, like the Sabbath law. However, we must not say, “Christians are under the law of tithing” if by that we mean that they are obligated to tithe in order to be righteous and holy before God. That’s legalism.
God has given Christians the wonderful gift of tithing. It will help us remember that all belongs to God; it will help us establish order in our finances; it will help us support the church; it will help us prosper. We do not have to tithe to be righteous. But, having been made righteous, shouldn’t we want our finances to align with biblical principles?