What does it mean that "many parts which are plain have been taken away"? The question https://christianity.stackexchange.com/questions/35704/to-what-point-in-time-can-the-bible-be-proven-accurate made me wonder: Is the common interpretation of 1 Nephi 13:26-28 that something was physically omitted from scripture? I am aware of the Joseph Smith Translation, but is the point to bring back physically removed parts? In at least one instance I know of, the "translation" is just a commentary, with no changed text, that makes it clear what is meant. This makes me wonder whether the goal was a corrected understanding, not so much a corrected text.

An other interpretation I can come up with is that, without requiring much change to scripture, doctrines that are introduced (e.g. Trinity) became basis for further theology, scripture translation (translation is always also in part interpretation) and so on. That also "[take] away from the gospel of the Lamb many parts which are plain and most precious".

So, is the official position as of now that there were things physically omitted from scripture, or that the cause is rather false doctrines themselves that became the spiritual basis for later generations? Or is there no "official" interpretation?
+1 for the great question! To begin as Members of the LDS Church "We believe the Bible to be the word of God as far as it is translated correctly; we also believe the Book of Mormon to be the word of God." (Article of Faith 8)

Throughout my response I will be referencing Plain and Precious Truths Restored, an article written in October of 2006.

There are multiple doctrines restored with The Book of Mormon. Just to list a few:


Premortal Existence
Adam’s Fall and Human Suffering
Agency
The Atonement
First Principles and Ordinances
Church Organization
Importance of Personal Revelation
Satan’s Identity and Methods


One example of a Joseph Smith Translation is Matthew 18:11

KJV: For the Son of man is come to save that which was lost.
JST of KJV: For the Son of man is come to save that which was lost, and to call sinners to 
     repentance; but these little ones have no need of repentance, and I will save them.


So in answer to your question, yes. We believe there have been doctrines which have been physically removed from The Bible either by an honest mistake in translation, or a deliberate omission.

-- I am out of time right now to go into more depth, but I would encourage you to read the article I referenced above.