And Melchizedek king of Salem brought forth bread and wine: and he was the priest of the most high God. And he blessed him, and said, Blessed be Abram of the most high God, possessor of heaven and earth: And blessed be the most high God, which hath delivered thine enemies into thy hand. And he gave him tithes of all.
Genesis 14:18-20 (KJV)

This Melchizedek was king of Salem and priest of God Most High. He met Abraham returning from the defeat of the kings and blessed him, and Abraham gave him a tenth of everything. First, his name means "king of righteousness"; then also, "king of Salem" means "king of peace."
Hebrews 7:1-2 (NIV)

Now the law requires the descendants of Levi who become priests to collect a tenth from the people-- that is, their brothers-- even though their brothers are descended from Abraham. This man, however, did not trace his descent from Levi, yet he collected a tenth from Abraham and blessed him who had the promises.
Hebrews 7:5-6 (NIV)

One might even say that Levi, who collects the tenth, paid the tenth through Abraham, because when Melchizedek met Abraham, Levi was still in the body of his ancestor.
Hebrews 7:9-10 (NIV)

Therefore, the Melchizedek Priesthood supercedes the Aaronic Priesthood.

Jesus is referred to as a High Priest after the order of Melchizedek. Jesus' Priesthood is not connected to Aaron, but to Melchizedek and is greater than Melchizedek because Jesus is God.  The Apostle Paul discusses this in Hebrews 7:2-3, 7:11-17, and 7:21:

...and Abraham gave him a tenth of everything. First, his name means "king of righteousness"; then also, "king of Salem" means "king of peace." Without father or mother, without genealogy, without beginning of days or end of life, like the Son of God he remains a priest forever.
Hebrews 7:2-3 (NIV)

If perfection could have been attained through the Levitical priesthood (for on the basis of it the law was given to the people), why was there still need for another priest to come-- one in the order of Melchizedek, not in the order of Aaron? For when there is a change of the priesthood, there must also be a change of the law. He of whom these things are said belonged to a different tribe, and no one from that tribe has ever served at the altar. For it is clear that our Lord descended from Judah, and in regard to that tribe Moses said nothing about priests. And what we have said is even more clear if another priest like Melchizedek appears, one who has become a priest not on the basis of a regulation as to his ancestry but on the basis of the power of an indestructible life. For it is declared: "You are a priest forever, in the order of Melchizedek."
Hebrews 7:11-17 (NIV)

but he became a priest with an oath when God said to him: "The Lord has sworn and will not change his mind: 'You are a priest forever.'"
Hebrews 7:21 (NIV)

In Summary:

1. Aaron's Priesthood (The Levites) were the keepers of the covenant of the Law. (Numbers 1:50-53)

2. Jesus' Priesthood is not connected to Aaron, but to Melchizedek and is greater than Melchizedek because Jesus is God.

Therefore, our tithe is not based upon keeping any part of the Law, but based on a precedence set forth in Genesis 14:18-20 and further emphasized in Hebrews 7.