This is a great question.
I'm not aware of any text from the Qur'an or hadith that explicitly mentions the gospel of Barnabas. Nor am I aware of any classical scholar who has addressed that gospel (but this is more than likely just my ignorance). As for modern positions, I don't know of any apart from the position taken by Yusuf Estes here and here:
But that is all nonsense, because the so-called "Gospel of Barnabas" is "Bogabas" (fake). The translators of this old manuscript were not convinced of any authenticity within it.
Most of the content deals with the mission of Jesus and his message and is quite compatible with all four gospels, another reason to doubt authenticity. Although the synoptic gospels are quite similar in many aspects, scholars have always insisted on Mark being the oldest, and most likely more authentic, and even then Mark is said to have been taken from another, much older document referred to only as "Q". The gospel of St. John is nothing like the others and is held to be in a class by itself by scholars. So, this begs the question; "How did a document all of a sudden surface 1500 years later, in Spanish (just after the time of the Catholic conquest of Muslim Spain in January of 1492) that brings together a melding together of all the gospels and the Quran and offers the same Islamic interpretation of Christian origins, all at the same time?"
Some scholars and academics from both Christians and Muslims have considered this gospel to be late and pseud epigraphical - But, others say it may contain remnants of earlier apocryphal (hidden from the public) work that was likely edited to conform to Islamic teachings, maybe from Gnostic or Ebionite or perhaps Diatessaronic sources. Some Muslim scholars have thought it to be a surviving version of a hidden apostolic original. Some Muslim academics have referred to it in support of their view of Jesus.
There are other works, not related but with similar names and are associated with apocryphal writings. These include the surviving Epistle of Baranabas and Acts of Barabas and even an earlier (unrelated) "Gospel" of Barnabas.
Although there are many "theories" being offered about this document, I seriously doubt it has the necessary credentials to be considered anything more than a very old manuscript that was translated by two people, Lon and Laura Ragg, along with their comments about 100 years ago.
Yusuf Estes is a former Christian preacher who became Muslim a while ago. Today he a popular Muslim preacher.
Frankly, I don't think many other serious scholars will take a position on this because it is fruitless. If one takes a position now, a few years down the line the authenticity might be disproven, or the text might be shown to have changed, or anything along those lines. Yusuf Estes is unique in that he is studied in the Christian tradition, so he may have the expertise to formulate an opinion.
Finally, the official position in Islam about Christian and Jewish texts (narratives on historical events, to be precise) are to neither affirm them nor to deny them. So for example the Bible has much more detailed narrations of historical events, and when we read those details, the Prophet (pbuh) told us to neither affirm them nor to deny them. I imagine a similar principle might apply in this case.
A reason that someone might say Islam does have a position on this is that in several places the Qur'an quotes Jesus and Moses (peace be upon them) as announcing the good news of another messenger (e.g. 61:6) and if I remember correctly, another place about this being recorded in the Books as well. And of course the Qur'an insists that the Books in their original form contained the message of monotheism and not any message deviating from that. Now whether the Gospel of Barnabas corresponds to the text that the Qur'an refers to, we don't know (and I argue, we cannot know until the Day of Judgment).
