It’s important that we not confuse the gospel of Barnabas (ca. A.D. 1500) with the Epistle of Barnabas (ca. A.D. 70–90). The Epistle of Barnabas was written in the late 1st century, but not by the Barnabas named in the New Testament. While more of a pseudo-Gospel with some historical value, the Epistle of Barnabas was never considered canonical by the early church or any church council.
The gospel of Barnabas, however, has absolutely no apostolic support and was written 1400 years after the time of Barnabas. This is evidenced by the fact that it was never quoted by any church father or church historian before the 16th century!
By contrast, the books of the New Testament were all written early (before A.D. 100) and by eyewitnesses, or by those who interviewed the eyewitnesses of the Lord Jesus (1 John 1:1-5; Luke 1:1-4). The four Gospels found in the New Testament were never questioned as to their authenticity.
Even had the gospel of Barnabas been written during the time of the Apostles, it still would have never attained canonical status due to the historical and doctrinal errors it contains. For example, the gospel of Barnabas purports that Jesus did not claim to be the Messiah (Matthew 26:63-64). The gospel of Barnabas also says that Jesus was born when Pilate was governor (but history records Pilate becoming governor in A.D. 26 or 27).
Furthermore, the gospel of Barnabas keeps some strange company, for it’s a favorite among Muslims as it teaches a Jesus consistent with the Koran. The gospel of Barnabas claims that Jesus did not die on the cross, as does the Koran in Sura 4:157. Historians are unanimous that the gospel of Barnabas was written in the 15th-16th century A.D., most likely by Muslims seeking to discredit the Biblical message regarding Jesus.