I am going to attempt to walk through the major literature in this discussion, which will be a lot of back and forth. I have linked to all the major works referenced, however not all of the articles and books are freely available online (some must be purchased).
Both Gordon D. Fee and Philip B. Payne are notable scholars who believe that 1 Corinthians 14:34-35 is an interpolation and should be omitted from the Bible. Other scholars who hold this position include Straatman, Fitzer, Barrett, and Ruef. In his article entitled Fuldensis, Sigla For Variants in Vaticanus, and 1 Cor 14.34–5, Payne points out that
"...scribes in that period simply did not take the liberty to rearrange the argument of Scripture in this manner. We do not have even a single parallel example of a scribe rearranging the sequence of an original text of any of the NT letters to make it more logical. Furthermore, even if Bishop Victor felt he had the authority to rearrange the sequence of the text, there is no adequate reason why the text would make more sense reinserted at the end of the chapter."
In his article MS. 88 as Evidence for a Text Without 1 Cor 14.34–5, Payne claims that
"The evidence that ms. 88 was copied from a text of 1 Corinthians 14 without vv. 34–5 provides additional external support for the thesis that vv. 34–5 were not in the original text of 1 Corinthians 14."
Curt Niccum wrote an article entitled The Voice of the Manuscripts on the Silence of Women: The External Evidence for 1 Cor 14.34–5 that claimed the bar-umlauts themselves were added to the text at a later date and are thus not indicative of an interpolation, but rather of a paragraphos (marginal note).
Payne co-authored an article with Paul Canart entitled The Originality of Text-Critical Symbols in Codex Vaticanus where they argue their point based on new findings concerning the ink used in the Codex; from the introduction to the paper:
"The discovery that the ink of text-critical symbols in Codex Vaticanus matches the original ink of the codex breaks new ground for textual criticism. A scribe in the Middle Ages, apparently concerned with fading, traced over the original ink of every letter or word of Vaticanus unless it appeared to be incorrect. Thus, unreinforced letters and symbols reveal the original ink of the codex. The most obvious examples of the original ink are the few places its scribe inadvertently duplicated a word, phrase or clause. In these cases the reinforcer traced over only one of the duplicates, so the other reveals the original ink."
Payne and Canart make the case that this finding further proves that the original manuscript omitted 1 Corinthians 14:34-35.
Payne based some of his beliefs about these texts being an interpolation on the work of J. Edward Miller who had posited that various umlauts (distigmai) might have been an indication of scribal uncertainty concerning the authenticity of these passages. However, after Payne cited Miller's work, Miller wrote an article entitled Some Observations on the Text-Critical Function of the Umlauts in Vaticanus, with Special Attention to 1 Corinthians 14.34-35 to refute Payne's position, arguing that Payne misinterpreted his findings. Payne wrote a followup article in response to this entitled The Text-Critical Function of the Umlauts in Vaticanus, with Special Attention to 1 Corinthians 14.34-35: A Response to J. Edward Miller, where he makes the case that the bar-umlaut does indeed have a special meaning.
Further work was published in 2007 studying umlauts in Codex Vaticanus, although it is not specifically related to the text in question (but sheds light on it). Unfortunately the work was published in French (and I am not aware of an English translation).
In 2009, Payne published his book Man and Woman, One in Christ: An Exegetical and Theological Study of Paul's Letters which contains his most current thoughts and most compelling arguments for 1 Corinthians 14:34-35 to be considered an interpolation, along with his views on all of the other passages dealing with gender roles in the Paul's letters. The book is very scholarly and does a great job considering multiple viewpoints, employing both historical-grammatical interpretation and textual criticism to make his points.
Despite Payne and others scholars' findings concerning distigmai in the text, conservative scholars still insist that this is insufficient evidence that 1 Corinthians 14:34-35 is an interpolation. Daniel B. Wallace summarizes this perspective in an article entitled The Textual Problem of 1 Corinthians 14:34-35, where he argues that despite the variance in where verses 34-35 are placed in the text, they must have been a part of the original text because they exist in all of the early manuscripts. Wallace makes the case that the Apostle Paul himself added the paragraphos.
In summary, there has been significant research conducted on both sides of this issue. Future work remains to be done concerning the meaning and dating of distigmai in manuscripts. Regardless of whether or not 1 Corinthians 14:34-35 is an interpolation, a parallel text exists in Paul's first letter to Timothy. This parallel text should always be considered when interpreting 1 Corinthians 14:34-35.