According to Didache chapter 13, the Gentile believers do not give the sacred gifts to priests and Levites but rather to prophets and teachers.
The Didache reasons that the “first part” should be given to the prophets because, as the writer indicates to the believers, “They are your high priests” (13.3). Some scholars interpret this as a replacement of the Levitical system, but the New Testament does not teach substitution or replacement of Old Testament laws and institutions, as Christian interpretation usually assumes. On the contrary, the New Testament assumes that the Torah’s laws of priestly gifts are being upheld.
Why did the Didache direct its readers to give their gifts to prophets and teachers rather than to Levitical priests? We should remember that the Didache was written to Gentile believers in Messiah who lived in the Diaspora. The laws of trumah in the Torah apply only within the land of Israel. Additionally, many of the communities in which the Gentile believers lived did not have qualified Levitical priests to whom trumah could be paid. Therefore, the Didache’s instructions are intended as special legislation for Gentile believers, teaching them how they may participate in the principles of the Torah’s priestly gift commandments. In this way the Didache reinterprets the trumah commandments of the Torah and makes them applicable to these new non-Jewish initiates.
It is interesting that we also find discussions in rabbinic literature about Gentiles and tithing. For example, we read in the Mishnah:
If a man carried his wheat to a miller who was a Cuthean or to a miller who was an am ha’aretz, [the wheat when ground continues] in its former condition in respect of tithes and the law of seventh-year produce. [but if he carried it] to a miller who was a gentile, [the wheat when ground becomes] demai. If a man left his fruit in the keeping of a Cuthean or of an am ha’aretz, [it continues when returned] in its former condition in respect of tithes and the law of seventh-year produce. [But if he left it] with a gentile, [it becomes] like the fruit of the gentile. Rabbi Simeon says: [It becomes] demai. (m.Demai 3:4)
The subject of this Mishnah is what happens to the status of tithed produce if it is given over to the custody of someone whose Torah observance is suspect. In other words, can they be trusted not to mix tithed produce with untithed produce and then sell it as all tithed? The Mishnah rules that Cutheans (a.k.a. Samaritans) and the am ha’aretz (non-observant Jews) can be trusted not to mix tithed and untithed produce but not Gentiles. Wheat given over to a Gentile miller is given the status of demai (doubtfully tithed produce) and fruit given over to a Gentile is given the status of the Gentile’s fruit, i.e., it needs to be tithed just like anything his fields produced.
What is interesting here is this relies on the opinion of Rabbi Meir who states that just because a Gentile acquires land within Israel, the produce from that land is still required to be tithed whether it is grown by a Jewish sharecropper or by the Gentile himself.  Elsewhere it was ruled that a Jewish sharecropper who transferred ownership of produce to the Gentile landowner did not have to separate firstfruits. 
Shemuel Safrai and Ze’ev Safrai argue that there was a debate going on as to Gentile obligations to tithing in the land of Israel:
The sages disagreed whether or not Gentiles were exempt from land-based commandments. In halachic literature, the decision is presented as a technically ideological issue. The technical debate was centered on the question if the separation of challah was incumbent upon the land itself, or on if the mitzvah was dependent on the worker of the land. The land is, of course, holy. However, the Gentile himself is exempt from the mitzvah. The ideological facet, however, is clear, does the holiness of the land stand on its own or not? 
There were all kinds of factors at play in these debates from trying to decipher a ruling from the text of the Torah itself to the advantage non-Jewish farmers within Israel would have in the marketplace if their produce was exempt from tithing. Shemuel and Ze'ev Safrai are convinced that in the Tannaitic Era the sages saw tithing as an obligation incumbent upon Gentiles whether or not they sold the produce to Jews.  However, it seems that most of the sages did not imagine that Gentiles themselves would tithe on the produce but that Jews buying produce from them would be required to tithe it themselves. This is not so much about figuring out what is required of Gentiles or how they would participate in the tithing system of the Torah as it is ensuring that Jewish people within Israel don’t eat from produce that hasn’t been tithed. It is within the context of these debates that we find the instructions of the Didache.
While these halachic debates and decisions have their importance, the Didache takes the discussion in a different direction. In the spirit of Matthew 16:19, “I will give you the keys of the kingdom of heaven, and whatever you bind on earth shall be bound in heaven, and whatever you loose on earth shall be loosed in heaven,” the Didache reinterprets and adapts the trumah commandments of Judaism to make them applicable to Gentiles living outside the land of Israel. The Didache’s main focus is not the details of halachah per se but finding a way for Gentile disciples to give back from the good gifts that their Father in heaven had provided for them.
For more on the Didache check out First Fruits of Zion’s new Messianic Jewish translation and commentary on the Didache entitled The Way of Life: The Rediscovered Teachings of the Twelve Jewish Apostles to the Gentiles.
- m.Bikkurim 1:2 and the discussion in Shmuel Safrai and Ze’ev Safrai, Mishnat Eretz Israel: Tractates Moed Katan and Hagigah with Historical and Sociological Commentary (Jerusalem, Israel: Bar-Illan University Press, 2012): 127-129.
- Pinhas Kehati, Mishnah: Seder Zera’im Volume 1 (trans. Rabbi Nahman Kahana; Jerusalem, Israel: Maro Wallace Press, 1994), 27.
- Translation by Jeremiah Detwiler from Safrai and Safrai, Mishnat Eretz Israel: Tractates Moed Katan and Hagigah with Historical and Sociological Commentary, 127-129.