There is no biblical instruction to refer to Jesus by the Aramaic or Hebrew pronunciation of his name. The New Testament is written in Greek and Jesus is always simply referred to as Ἰησοῦς (Iēsous), the Greek spelling of Joshua from the Old Testament.
In fact the name Jesus in the New Testament can refer to five people:
1. Jesus of Nazareth
2. Jesus Barabbas the criminal whom Pilate released instead of Christ
3. Jesus, meaning Joshua, Moses’ successor (Acts 7:45, Hebrews 4:8 in Greek text and KJV)
4. Jesus son of Eliezer, one of the ancestors of Christ (Luke 3:29)
5. Jesus Justus, one of Paul’s co-workers in the gospel (Colossians 4:11)
Yeshu, Yeshua, Yehoshua?
We also don’t really know the pronunciation of Jesus’ name in Aramaic, the colloquial language of the time – some scholars think it would have been Y’shu’, some Y’shua . In formal terms, as a given Hebrew birth name, the name was correctly pronounced Yehoshua, the same as the successor of Moses.
As early as John Kitto (1861) various scholars had given ‘Yeshua’ as an Aramaic form of Jesus, but modern use in a small number of churches is generally associated with Christian groups attempting to ‘restore’ (or arguably introduce) Jewish elements into Christian practice, such as keeping Jewish feasts or observing the Friday evening to Saturday afternoon Sabbath. Sometimes these groups are Messianic Trinitarian churches, sometimes they are Arian churches (churches which believe that Jesus existed before his birth but was not himself God).
The usage sometimes is accompanied by the belief that the Hebrew language is itself sacred or more meaningful and should be retained in names. Unfortunately the only example in the New Testament of anyone reverting to an Aramaic rather than Greek name is the case of Simon Peter (literally in Greek, petros, Simon the Stone) who in Antioch in the incident where he was influenced to not break bread with Gentile Christians reverted to the name Cephas (kefa, stone in Aramaic, Galatians 2:11). The Aramaic name Cephas was then taken up by the conservative Jewish faction in the division of the church in Corinth: “I am of Cephas” (1 Corinthians 1:12). So this is not a good example of using an Aramaic name, in fact it seems to have a been a sign of divisive adherence to Jewish practices.
The main reason not using the name Yeshua would simply be that Jesus was to be “like his brothers in all ways” (Hebrews 2:17), so to have just Jesus named with a Hebrew name, but his brothers – the disciples – named using normal Greek or English Bible names – John, James, Andrew, Peter and so on – makes Jesus stick out strangely in a way the New Testament writers never intended. It is distorting the Bible text for effect.
The objection may come “But calling Jesus Yeshua shows the link to Yahweh”. The problem here is that the New Testament, and Jesus himself when reading in the synagogue or quoting the Old Testament never actually uses the name “Yahweh”, so why do some Christians? True the reasons that Jews did not use the name in Jesus’ time are themselves weak. The pronunciation out loud of the divine name YHWH as “Yahweh” (or in Hebrew “Y’hua”) did not become taboo until after the return from Babylon, probably during the time of the Maccabees. Nevertheless it was taboo during Jesus’ day and it appears to be a taboo that he always respected. As did the writers of the New Testament.
The other problem is that Jesus (Yehoshua = “Yah saves”) is not the only name in the New Testament related to the name of God. Other names include John (Yehannan, = “Yah gives”), Matthew (Mattathias = “gift of Yah”) among many others. Giving Jesus a non-Greek non-English makes him stand out of from the other names of the New Testament, including the four other Jesus’s, but it isn’t something there in the original text.