The Tetragrammaton does not occur in any extant Greek manuscripts of the New Testament. Extant Greek New Testament manuscripts contain the Greek word Kyrios (Lord) in Old Testament quotes where the Hebrew has the Tetragrammaton.
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Manuscripts
None of the extant Greek manuscripts of the New Testament contain the Tetragrammaton. The oldest New Testament manuscripts, P52, and P104 cannot be used in discussions because their text doesn't include any verses in which the divine name appears, although Gerard Gertoux mentions the P52 by the absence of Nomina Sacra.
A passage recorded in the Hebrew Tosefta, Shabbat 13:5, quoting Tarfon is sometimes cited to suggest that early Christian writings or copies contained the Tetragrammaton.
Shabbat 13:5
This same source quotes Rabbi Jose the Galilean (who lived in the 1st and 2nd centuries of the common era): “one cuts out the references to the Divine Name which are in them [the Christian writings] and stores them away, and the rest burns.”
Laurence Schiffman views this as a discussion of whether to rescue section of the sifre minim (Hebrew language texts of Jewish Christians) containing the tetragrammata from a house fire. Another interpretation suggests this is a reference to Old Testament Torah and not the Gospels.
Although none of the extant Greek New Testament manuscripts contain the Tetragrammaton, scholar George Howard has suggested that the Tetragrammaton appeared in the original New Testament autographs, and that "the removal of the Tetragrammaton from the New Testament and its replacement with the surrogates κυριος and θεος blurred the original distinction between the Lord God and the Lord Christ." In the Anchor Bible Dictionary, Howard states: "There is some evidence that the Tetragrammaton, the Divine Name, Yahweh, appeared in some or all of the OT quotations in the NT when the NT documents were first penned."
Along with Howard, David Trobisch and Rolf Furuli both have suggested that the Tetragrammaton may have been removed from the Greek manuscripts. In the book Archaeology and the New Testament, John McRay wrote of the possibility that the New Testament autographs may have retained the divine name in quotations from the Old Testament. Robert Baker Girdlestone stated in 1871 that if the Septuagint had used "one Greek word for Jehovah and another for Adonai, such usage would doubtless have been retained in the discourses and arguments of the N.T. Thus our Lord in quoting the 110th Psalm,...might have said 'Jehovah said unto Adoni.'" Since Girdlestone's time it has been shown that the Septuagint contained the Tetragrammaton, but that it was removed in later editions. For example the 8HevXII gr manuscript dated to the 1st century CE contains the Tetragrammaton in Hebrew or paleo-Hebrew script.
Wolfgang Feneberg comments in the Jesuit magazine Entschluss/Offen (April 1985): "He [Jesus] did not withhold his father's name YHWH from us, but he entrusted us with it. It is otherwise inexplicable why the first petition of the Lord's Prayer should read: 'May your name be sanctified!'" Feneberg further notes that "in pre-Christian manuscripts for Greek-speaking Jews, God's name was not paraphrased with kýrios [Lord], but was written in the tetragram form in Hebrew or archaic Hebrew characters. . . . We find recollections of the name in the writings of the Church Fathers".
Though Albert Pietersma, along with most scholars, does not accept Howard's theory, Pietersma has stated concerning the Septuagint: "It might possibly still be debated whether perhaps the Palestinian copies with which the NT authors were familiar read some form of the tetragram."
Tatian's Diatessaron shows some variance in applying Kyrios to YHWH, but this may be because of dependence on the Peshitta. The consistency in rendering YHWH as Kyrios in all New Testament references would be difficult to explain if there were not already either an established tradition to read Kyrios where YHWH appears in a Greek manuscript, or an established body of texts with Kyrios already in the Greek.
Hebrew translations
Over the centuries, various translators have inserted the Tetragrammaton into Hebrew versions of the New Testament. One of the earliest Rabbinical translations of Matthew is mixed in with the 1385 critical commentary of Shem-Tob. He includes the Tetragrammaton written out or abbreviated 19 times, while occasionally including the appellative HaShem (השם, meaning "The Name").
English translations
Most English Bibles, even those such as the Jerusalem Bible which has Yahweh in the Old Testament, do not use Yahweh in the New Testament. This is because the Greek New Testament manuscripts are quoting the Septuagint, where the Hebrew YHWH is translated as kyrios. The New Testament uses Greek kyrios for YHWH even, for example, when Jesus reads the Isaiah scroll at the synagogue in Nazareth (Luke 4:17–19 reading Isaiah 61:1).
A few English translations of the Bible do use Jehovah in the New Testament. For example, William Newcome, in what is sometimes known as "Archbishop Newcome's new translation", has the name Jehovah a few times where the New Testament quotes from the Old Testament, such as Matthew 22:24. The first complete Bible printed in America by John Eliot, although not in English, frequently uses "Jehovah" in the New Testament.
The New World Translation
The rendering Jehovah appears 237 times in the New World Translation (NWT) of the New Testament published by Watch Tower Bible and Tract Society and used by Jehovah's Witnesses. Jehovah's Witnesses say that the authors of the New Testament writings retained the Tetragrammaton in their quotations of the Old Testament without substituting it with Kurios.
Sacred Name Bibles
In 1993, the Institute for Scripture Research (ISR) published The Scriptures, the first English translation to incorporate the Hebrew letters of the Tetragrammaton instead of a generic title (e.g., the LORD) or a conjectural transliteration (e.g., Yahweh or Jehovah). The Besorah and ISR's The Scriptures '98 also incorporate the Tetragrammaton, using Paleo-Hebrew script rather than Hebrew square script.
The Divine Name King James Bible, published in 2011, uses Jehovah and Jah in 7,023 places where the capitalized LORD and GOD appeared in the Authorized King James Version. Jehovah appears in parentheses in the New Testament portion of this Bible where Old Testament quotes are cross-referenced.
The Sacred Scriptures Bethel Edition (1981) renders the Tetragrammaton as Yahweh in both the Old and New Testaments.
The Emphatic Diaglott (1864), by Benjamin Wilson, renders the divine name as Jehovah in eighteen instances in the New Testament.
The Original Aramaic Bible in Plain English (2010) by David Bauscher, a self-published English translation of the New Testament, from the Aramaic of The Peshitta New Testament with a translation of the ancient Aramaic Peshitta version of Psalms & Proverbs, contains the word "JEHOVAH" approximately 239 times in the New Testament, where the Peshitta itself does not. In addition, "Jehovah" also appears 695 times in the Psalms and 87 times in Proverbs, totaling 1,021 instances.