Many Byzantine priority/majority text/textus receptus advocates rely on Dean John W. Burgon’s massive collation of patristic evidence a century ago. Dean Burgon found over 85,000 quotations in the early fathers that he said used the Byzantine text. But Burgon used uncritical and late texts (copied in the middle ages) and made a number of assumptions about the fathers when they quoted the NT (for example, Ignatius and Irenaeus often wrote, ‘As the Lord said,’ without giving the book name. Burgon found the wording in Mark that was Byzantine—though the wording in Matthew was Alexandrian—and he then assumed that the patristic writer was quoting from Mark). This issue has been raised by numerous scholars over the years. Gordon D. Fee, who is probably the best patristic text-critical scholar alive today, has said that there are NO ante-Nicene fathers who quoted the Byzantine text. As well, there is a recent article in the Journal of the Evangelical Theological Society that deals just with Burgon’s approach. The author found that it was terribly faulty.
As for versional evidence, there used to be, about 70 years ago, the dispute as to whether the Syriac Peshitta was Byzantine or not. This was coupled with issues related to its date. Burgon’s followers dated the Peshitta in the second century and defended the view that it was Byzantine. Others argued that it was fifth century and was non-Byzantine. After all these years, we now can come to a better perception than either side had: the Syriac Peshitta is indeed fairly early, sometime before the year AD 451. It may even be as early as the third century. But its textual character is not Byzantine. The earliest Syriac translation that is Byzantine is the Harclean version of the sixth century.
Others claimed that the Gothic version of the late fourth century was the earliest Byzantine text. But recent work in the Gothic version suggests that it is not Byzantine. I don’t think the work done is yet adequate to make such a claim, so I am not willing to entirely abandon the view that the Gothic may be the earliest Byzantine version. Nevertheless, it is significant that the more research that is done on the versions and fathers the less they look Byzantine.
The evidence is rather overwhelming. There are a few folks who would claim that the Byzantine text existed early in the versions and fathers, but their methods are flawed and they represent no more than about 1-2% of all textual scholars. But even if they could prove that the Byzantine text was early, this would not be enough: they would also have to demonstrate that it was the predominant text-form in the early centuries. On the other side, there are the vast hordes of textual scholars of all theological stripes who see no real evidence that the Byzantine text was early. Usually the argument against these scholars in fact turns on their convictions. You will notice that Byzantine/KJV folks argue along two lines almost all the time: God has preserved his text and since the Byzantine is the most amply preserved, it must go back to the original; and the scholars who are behind modern translations are either deceived or are themselves heretical. Thus, their arguments are anything but rational; they are usually emotional and ad hominem. Frequently, Westcott and Hort are maligned as liberal and heretical, even as occult leaders (a charge that is blatantly false). Two things are conveniently overlooked when such ad hominem charges are made: first, the textus receptus (which was the Greek text used by the KJV translators) was produced by a Roman Catholic humanist who, by the standards of the KJV advocates, should be labeled as far more heretical than either Westcott or Hort; and second, the character of men like Hort or Westcott really has nothing to do with our evaluation of the ancient evidence, nor does Erasmus’ character. No textual scholar today completely follows Hort’s approach; at the same time, the great mass of evidence found in the last century largely confirms his general direction and has certainly done nothing to give comfort to KJV advocates.