Who first distinguished between the inerrancy of the Bible and the inerrancy of the original autographs? It's common these days among those who believe in the inerrancy* of the Bible to limit the statement to the original manuscripts.  For example, the Chicago Statement reads:


  We affirm that inspiration, strictly speaking, applies only to the autographic text of Scripture [...]. We further affirm that copies and translations of Scripture are the Word of God to the extent that they faithfully represent the original.


Augustine apparently saw matters similarly:


  If we are perplexed by an apparent contradiction in Scripture, it is not allowable to say, The author of this book is mistaken; but either the manuscript is faulty, or the translation is wrong, or you have not understood. (Contra Faustum, 11.5)


Who is the first Christian to specifically distinguish between the inerrancy of the copied and/or translated biblical text, and the inerrancy of the original autographs?

* Note that I'm defining "inerrancy" broadly, so that it includes both Evangelical and Catholic understandings of the doctrine.
In an essay titled "Inerrancy as Inheritance: Competing Genealogies of Biblical Authority," Published in a book titled Evangelicals and Scripture: Tradition, Authority and Hermeneutics, a collection of essays by various authors, Dr. Thomas Buchan, Associate Professor of Church History at Nashotah House Theological Seminary  reviews the late Dr. Harold Lindsell's The Battle for the Bible, Dr. Jack  Rogers and Dr. Donald McKim's The Authority and Interpretation of the Bible: An Historical Approach, and Dr. John Woodbridge's Biblical Authority: A Critique of the Rogers/McKim Proposal. In this essay, Dr Buchan notes, that Lindsell, Rogers and McKim all agree that the doctrines of biblical inerrancy were not meaningfully discussed or engaged with by theologians until after the 19th centry.


  Lindsell's work, in my opinion, correctly identified the fact that the issue of biblical inerrancy did not become a matter of contention until the nineteenth and twentieth centrues.
  
  ...
  
  Rogers and McKim argued correctly that biblical inerrancy as he conceived of it could not be found in the earlier periods of church history.


Dr. Buchan goes on to note:


  These three works sought to settle the question of the status of biblical inerrancy as the authentically evangelical doctrine of Scripture by means of accounting for its historical lineage. Our consideration of them ,however, raises another question: does the project of writing the history of biblical authority need to be re-envisioned? We must face the possibility that Lindsell was right when he observed that the inerrancy of the bible has only become a controversial issue within the recent history of the church. If this is so, we ought not be surprised if we fail to find inerrancy as it has been articulated in the twentieth-century evangelicalism in the theology of the patristic, medieval or Reformation periods.


And concludes by saying,


  None of this rules out a an evangelical commitment to biblical inerrancy as a theology of Scripture, but it does call into question the perceived weight of the claim that an evangelical conception of biblical authority is historically normative.


This is not to say that the doctrine of inerrancy did not exist as an assumed fact by early church fathers, as other answers, like Dick Harfield's have pointed out. The problem is that because the inerrancy of scriptures was assumed, little to no meaningful thought was given to the difference between the original autographs and present manuscripts. The assumption was that there was no difference or only minor topographical errors - errors so minor as to render them essentially meaningless differences. As Mr. Harfield notes:


  Church Fathers up to the time of Augustine were unanimous in considering the scriptures perfect and inerrant, with no apparent thought that the manuscripts known to them differed in any way from the autographs.


Within Jewish circles, there was a belief that scripture was inerrant as early as 200CE (or as late as 500CE) as recorded in the Babylonian Talmud. In A History of the Jewish People in the Time of Jesus Christ, author Emil Schürer notes


  
    "He who says that Moses wrote even one verse of his own knowledge is a denier and despiser of the word of God"3
  
  
  The whole Pentateuch was thus now regarded as dictated by God, as prompted by the Spirit of God.4 Even the last eight verses of Deuteronomy, in which the death of Moses is related, were said to have been written by Moses himself by means of divine revelation.5 Nay at last, the view of a divine dictation was no longer sufficient. The complete book of the law was declared to have been handed to Moses by God, and it was only disputed whether God delivered the whole Thorah to Moses at once or by volumes.6
  
  3 Babylonian Talmud Sanhedrin 99a
  4 See in general, Joh. Delitzsch, De inspiratione scriptura sacrae quid statuerint patres apostolici et apologetae secundi saeculi (Lips, 1872, pp. 4-8, 14-17
  5 Baba bathra 15a (lat. in Marx, Traditio rabbinorum veterrima de librorum Bet. Test ordine atque arigine, Lips. 1884, p 23). Philo, Vita Mosis iii. 39 ed Mang. ii 179). Joseph. anit. iv. 8 48.
  6 Gittin 60a


Yet despite this, there was still no distinction between the textis receptus and the manuscripts of the time. This was largely due to the belief that not just the written Torah was inerrant, but the oral Torah was also inerrant and supernaturally preserved as well. User [Aaron Shaffier] of the Mi Yodea Stack Exchange explains:


  Maimonides (also know as the Rambam) codifies 13 principles which are basic to Judaism. These principles are pretty much universally accepted as binding in all Orthodox forms of Judaism. Principle number 8 is, "The belief in the divine origin of the Torah." Principle number 9 is, "The belief in the immutability of the Torah."
  
  This being said, it is important to differentiate between the idea of the Torah being perfect as understood in Judaism and the concept of "Biblical Inerrancy" as understood by many Christian groups.
  
  Judaism believes that the Torah was revealed in two parts. The written text of the Torah was dictated to Moses exactly as we have it today. Together with this "Written Torah", much additional information about each commandment, as well as a complete system of Torah interpretation was was also given to Moses and passed down through the generations.
  
  After the destruction of the Second Holy Temple in 70 CE, a process of collecting and recording these teachings was begun. The teachings of the Oral Torah were eventually codified in the Talmud, Midrash, Zohar etc. This process is explained in great detail in the Introduction of the Mishneh Torah by Maimonides.
  
  This means that according to Judaism, one cannot just read the text of the Written Torah and fully understand what God wants from us. One needs to also consult the teachings of the Oral Torah.


This explanation, which is supported by Dr. Elizabeth Shanks Alexander in her essay "The Orality of Rabbinic Writing" published in The Cambridge Companion to the Talmud and William Gaventa in his book Jewish Perspectives on Theology and the Human Experience of Disability provides an important clue as to why, though church fathers may have mentioned that there could be differences between the manuscripts of the time and the original autographs, this did not rise to the level of being a distinction between the modern manuscript being inerrant and the original autographs being inerrant.

In much the same way that Judaism had a tradition of oral Torah and a system of adjudicating and clarifying apparent errors in the Torah, so too did Catholicism. Should an issue or question regarding the inerrancy the the original autographs vs. the Vulgate, Geneva Bible or King James, one need only look to the Pope for clarification through an encyclical, decretal, Papal Bull, or other such means of issuing church doctrine. This rendered the question of inerrancy and the whole discussion moot and rather silent until it began to stir in 1512 when Desiderius Erasmus began to collect as many variants and copies of the Vulgate as he could in order to publish a Greek and Latin New Testament Critical Text. Shortly thereafter (1645–1707) John Mill collated 82 additional Greek manuscripts into the Novum Testamentum Graecum, cum lectionibus variantibus MSS.  he reprinted Desiderius Erasmus' original work, but also enumerated 30,000 textual variants. A few years after publication, Daniel Whitby criticized the work of Mill, and in some way it could be said that this was the first inkling of the idea that only the original autographs might be inerrant. About this same time, Richard Simon, a French Catholic Priest who published a work titled A Critical History of the Old Testament in 1682. This work was controversial in the Catholic Church and 1,300 copies of his work were seized and destroyed by royal decree. According to historian Philip Shaff in Theological Propaedeutic: A General Introduction to the Study of Theology it was Simon who was first to make the distinction between the inerrancy of the Bible and the inerrancy of the original autographs in a meaningful way.

It was not until 1881 with the publication of Inspiration by Archibald Hodge and Benjamin B. Warfield, leaders of Princeton Theology that the issue of Biblical inerrancy entered the mind of the theology community, and Warfield and Hodge put forth the doctrine that it was only the original autographs which were inerrant. Only after the topic of inerrancy entered the main stage of theological discussion could the more narrow idea of the inerrancy of the original autographs be put forth and discussed. Though it is obvious that they were not the first, Warfield and Hodges were the most influential in bringing this idea to center stage and it was Inspiration which introduced the topic to the world as we know it today.

This also coincides with the rise of archaeology at the beginning of the 1800s and the Discovery of the Dead Sea Scrolls discovered between 1946 and 1956. As we began to discover older manuscripts and were able to compare them, only then were we able to realize just how different the original manuscripts could be from the present day manuscripts, thus bringing into focus the idea and question that it might only be the original autographs which were inerrant.

Software developer Brandon Staggs has an excellent video discussing the topic.