Where did the claim that "the Bible we have today is unreliable because of retranslation" come from? I've heard many people assert that the Bible we have today is unreliable because of a long chain of translation-of-a-translation. This seems false, because new translations can use ancient manuscripts as their source. What is the origin of this fallacy? Is there any one organization to blame in propagating it?

Here's a link to my basis for the question.
At the risk of being overly pedantic, we need to understand what Pastor Litke means by "reliable" and perhaps what he even means by "the Bible".

The Concise Oxford English Dictionary defines "rely" as (1) "trust fully; have faith in" and (2) "be dependent on".  But these definitions suppose an object:  to trust fully for what?  to depend on for what?  Do we expect, for example, that the Gospel accounts convey the literal meaning of what Jesus is purported to have said, or simply the sense of what he said?  Do we expect that the creation account(s) are scientifically accurate or in some sense allegorical?

In the article you linked, Pastor Litke seems to define "reliability" as the extent to which the true meaning of a particular verse or phrase is conveyed.  Paraphrases, he writes for example, are "less reliable, because you only know what the person doing the paraphrase thought a particular verb or phrase means."  He defines a paraphrase as:


  a less literal rendering of the Bible – restating the text to give the
  original sense but not attempting to literally translate each term in
  the original language.


He contrasts "paraphrases" with what he refers to as "translations":


  a rendering of the Bible in a language different than the one in which
  it was written ... intended to be as literal as possible and still be
  easily read


"Translations" of this sort, he maintains, are more "reliable" than paraphrases.  



The following is an interesting example to consider in the context of Pastor Litke's argument.

The Greek text (Textus Receptus) of John 5:27-28 reads:


  και εξουσιαν εδωκεν αυτω και κρισιν ποιειν οτι υιος ανθρωπου εστιν μη
  θαυμαζετε τουτο οτι ερχεται ωρα εν η παντες οι εν τοις μνημειοις
  ακουσονται της φωνης αυτου


The 1900 revision to the King James Version reads:


  And [the Father] hath given Him authority to execute judgment also, because He is the Son of Man. Marvel not at this


According to Pastor Litke's definition of reliability, this translation is reliable in the sense it conveys the literal meaning of the each phrase:


και εξουσιαν εδωκεν αυτω και κρισιν ποιειν literally means "and authority gave to him also judgment to execute"
οτι υιος ανθρωπου εστιν literally means "because son of man he is"
μη θαυμαζετε τουτο literally means "marvel not at this"


But, since the original manuscripts had no punctuation, these same words can be arranged to form a different set of phrases altogether:


  He hath given Him authority to execute judgment also.  That He is
  the Son of Man, marvel not at this


According to Pastor Litke's criteria, this also would be a "reliable" translation, since it conveys the literal meaning of the underlying phrases.

Theologically, however, there is a great gulf between these two different translations.  The ambiguity in the Greek was understood and addressed in antiquity by Byzantine Church Fathers, but seems to have been lost in most English translations.  Nor is it an ambiguity that can be addressed through a mechanical appeal to dictionaries, concordances and lexicons.  John Chrysostom recognized the pitfall in reading the verse incorrectly and highlit the problem in his 39th Homily on the Gospel According to John:


  That He is the Son of Man, marvel not at this
  
  Paul of Samosata rendereth it not so; but how? “Hath given Him
  authority to execute judgment, ‘because’ He is the Son of Man.” Now
  the passage thus read is inconsequent, for He did not receive judgment
  “because” He was man, (since then what hindered all men from being
  judges,) but because He is the Son of that Ineffable Essence,
  therefore is He Judge. So we must read, “That He is the Son of Man,
  marvel not at this.” For when what He said seemed to the hearers
  inconsistent, and they deemed Him nothing more than mere man, while
  His words were greater than suited man, yea, or even angel, and were
  proper to God only, to solve this objection He addeth,
  
  Marvel not [that He is the Son of Man] for the hour is coming in the which they that are in the tombs shall hear His voice and shall go
  forth, they that have done good to the resurrection of life, and they
  that have done evil to the resurrection of judgment.


As a result, I would say that Pastor Litke is correct in his assertion that the King James Version and the other translations he cites are "reliable", but only in the narrow sense he suggests. 

There is also a somewhat insipid issue regarding what "the Bible" actually is.  Why, for example, is the Gospel according to John in any of the versions that Pastor Litke cites in the first place?  If a particular version of the Bible contains a faithful translation of the Shepherd of Hermas, would that particular Bible be considered "reliable"?  Hermas was included, after all, in the Codex Sinaiticus, which is considered by scholars to be one of the most reliable manuscript sources and is either the oldest or second oldest complete codex we have of the New Testament.  

I think the problem with the basis of your question is that Pastor Litke's treatment of Biblical "reliability" is too superficial.  (I am not criticizing your question.  I personally think it is excellent.). The King James (and other) translations of John 5:27-28, are "reliable" in Pastor Litke's understanding, but convey a teaching (of Paul of Samasota) that was considered heretical by the Church in its 3rd century of existence (and is still considered heretical within the Roman Catholic and Eastern Orthodox Churches at least).  The Bible versions he lists exclude some books that were initially considered but later rejected by the Church for inclusion (e.g. Hermas) and exclude others that were prescribed for inclusion by the Church canons, but later excluded by reformers (e.g. some of the Deuterocanonical books).  Pastor Litke's framework for "reliability" doesn't accommodate differences of canon at all.