This claim, while repeated in various places, is actually a bit dubious.
The statement of almost all of it being quoted in the first three centuries is attributed to David Dalrymple. But it
doesn't seem to be in any of his works. Instead, the whole claim seems to come from a secondhand source saying this is something David Dalrymple said to him. This is mentioned in a footnote of "Memoirs of the Lives of Robert Haldane of Airthrey, and of His Brother, James Alexander Haldane" in a footnote on pages
481-482. Essentially, a conversation came up about how much of the New Testament could be recovered from quotations of the first three centuries. Two months later, one of the people involved (Dalrymple) said he felt he had found the answer, and is quoted as saying:
"There have I been busy for those two months, searching for chapters, half chapters, and sentences of the New Testament and have marked down what I found, and where I have found it, so that any person may examine and see for himself. I have actually discovered the whole New Testament, except seven or eleven verses (I forgot which), which satisfies me that I could discover them also."
(I want to admit I was not the one who found the above linked references; it was someone else who looked into it. However, the material below is my own work)
The problem here is that this is secondhand testimony and vague at that, as we do not actually know which these verses are, and it even says he isn't sure how many (it's unclear to me whether the "I forgot which" is Dalrymple saying he forgot, or the writer reporting it saying he forgot which he said). This claim could still be true, but unless we have the testimony of someone who can show their work on the subject--or at least
firsthand testimony of someone who did the work--the claim must be regarded as uncertain. Has there been anyone who can demonstrate that all but a few verses can be found in the first three centuries? I'm sure there must have been
someone who made something where they included a note for each New Testament verse as to who quoted it.
In order to try to determine it, I tried to make use of this site:
www.catholiccrossreference.online
This site is for looking up quotations of the church fathers of the Bible. You put in the verse, and it'll tell you the ones that quoted it. Now, it should be noted that this is only searching through the works found at CCEL and thus is missing any that weren't in those. However, I believe CCEL has pretty much all of the ones from the first three centuries (except Irenaeus's
Demonstration of the Apostolic Preaching, because a copy was only discovered in 1905, and the Ante-Nicene Fathers series), and it's later ones that it's missing... for example, none of Jerome's biblical commentaries are there. But they've gotten the first three century ones, so we can search. One other issue with it is that searches for footnotes, so a footnote mentioning a verse, even if it's not in reference to an actual quote, will still show up. It even triggers footnotes for introductory material that the editor included, and footnotes that say something like "see Matthew 2" even if only a small part of it is actually quoted mean any verse from Matthew 2 will find it. One also must in some cases confirm it isn't a
spurious writing; for example, epistles attributed to Ignatius that are stated to be spurious even in the Ante-Nicene Fathers series will be triggered and it won't clearly label them as such.
The first one listed for Matthew 1:1 is a quote by Ignatius... however, this is from one of the
spurious epistles (regrettably, the search does not clearly articulate this was in the "spurious" section). However, other writers from the first three centuries do refer to it, like Irenaeus, o we have a quote on this one. However, Matthew 1:2 through Matthew 1:11 is included in any early church father writing. Matthew 1:12 turns up a
reference, but not a direct quote, by Irenaeus ("For Joseph is shown to be the son of Joachim and Jechoniah, as also Matthew sets forth in his pedigree" which the footnote says refers to Matthew 1:12-16). I am not sure if a reference rather than a quote satisfies the requirement. Let's suppose it does. Moving forward, Matthew 1:17 and 1:18. 1:19 runs into a weird spot where we don't find quotes exactly from early church fathers, but we do see
apocryphal writings like the Gospel of the Nativity of Mary as well as the Diatessaron (a combination of the Gospels into a single document) say these things. I'll be generous and count them. So with these generous counts, we do get the rest of Matthew 1.
However, we see that in the first chapter of Matthew we find 10 verses not quoted, even with being generous in what we accept as a quote. Because the Diatessaron would cover so much of the gospels I decided to skip over to Acts, and we indeed find no quote of Acts 1:2 until Augustine, the same with Acts 1:13. I want to again stress I am trying to be as generous as possible as to what constitutes a quote; some of the ones I counted as being quoted were just references rather than an actual quote or were in some kind of apocryphal work or were just included the Diatessaron, a compilation rather than being a real quotation. Oh, and if anyone is wondering, I did search for "Acts" and "Matt" in Demonstration of the Apostolic Preaching (as it does mention when there are quotations) and the ones I mentioned did not turn up there either.
So here we have, being very generous with the definition of quotations, twelve verses already not found. And I didn't really bother to look through the rest of the gospels (including most of Matthew), the rest of Acts, any of the epistles, or Revelation, where I'm sure I could find more. The point here was merely to see if I could exceed 11, and I already did. So this claim of all but eleven--which was secondhand testimony--seems to be false, and yet this is the actual
source for the much-repeated "almost all of the NT was quoted" in the first three centuries.
tlr;dr version: The claim that "almost all of the New Testament was quoted by the church fathers of the first three centuries" appears to date back to a secondhand anecdote of someone claiming they found all but seven or eleven verses of the New Testament in the first three centuries (without specifying what those quotations were). But even using a generous definition of quotations, I was able to find (in the first chapter of Matthew and the beginning of the first chapter of Acts) twelve verses not quoted in that period; as I had already gone beyond eleven, I stopped there. So this talking point doesn't seem accurate. We can clearly find a lot of verses quoted in the first three centuries, but not almost all.