Q:

What is the status of the New Testament in the LDS teachings?

According to the LDS, are the New Testament scriptures- as they existed and were available at the commencement of that denomination- themselves incomplete or the result of apostasy, or was it the interpretation and application of those works among the Church that needed to be restored through a latter-day revelation? 

A:

The New Testament, though not viewed as the cornerstone of the LDS Doctrine or church, is the foremost authoritative document on the mortal life and ministry of the Lord Jesus Christ as well as the ministry of the Apostles in teaching the young church how to live the new or "Higher" law as taught by Jesus Christ. Speaking of the New Testament, the late Elder L. Tom Perry of the Quorum of the Twelve Apostles said this: 

"This sacred volume is the centerpiece of scriptural history, just as the Savior Himself should be the centerpiece of our lives. We must commit ourselves to study it and treasure it!" (The Sabbath and the Sacrament, May 2011 Ensign, 6)

ShemSeger is correct in that the translation is, at times, called into question and the Joseph Smith Translation was created to clarify the mistranslations, but that, by itself, is not the status of the New Testament in LDS Doctrine. The Joseph Smith Translation only serves as a study tool and does not replace the New Testament. 
As you look at the teachings of the leaders of the church in General Conference, modern revelation and their own published doctrinal books, you'll see that they are all consistent with this status. 
As Examples of published doctrinal books, Jesus the Christ By James E. Talmadge and Bruce R. McConkie's Messiah Series (consisting of The Promised Messiah, four volumes of The Mortal Messiah, and The Millennial Messiah)
It is the when the other books of scripture are read through the lens of the New Testament that we begin to understand the fullness of the Nature and Divinity of Jesus Christ. The Old Testament's main role is prophesying that Christ will come and atone for the sins of the world (Luke 24:44; John 5:39) and the Book of Mormon's main purpose is to act as a second witness and testament of Christ's divinity and mission and testifying to the validity of the Bible, both Old and New Testament (Mormon 7:9). When the prophecies and doctrines pertaining to Christ in these two other books of scripture are viewed through this lens of the New Testament, it helps us understand more of Christ's mission and ministry, but they, standing alone, are not enough and are not able to fulfill their mission without the New Testament's existence.
The coordination of these was mentioned by Elder M. Russell Ballard of the Quorum of the Twelve Apostles in April of 2007:

"The Book of Mormon does not dilute nor diminish nor de-emphasize the Bible. On the contrary, it expands, extends, and exalts it. The Book of Mormon testifies of the Bible, and both testify of Christ.
  The first testament of Christ is the Bible’s Old Testament, which predicted and prophesied of the coming of the Savior, His transcendent life, and His liberating Atonement.
  The second Bible testament of Christ is the New Testament, which records His birth, His life, His ministry, His gospel, His Church, His Atonement, and His Resurrection, as well as the testimonies of His Apostles.
  The third testament of Christ is the Book of Mormon, which also foretells Christ’s coming, confirms the Bible’s account of His saving Atonement, and then reveals the resurrected Lord’s visit to the earth’s other hemisphere. The subtitle of the Book of Mormon, the clarifying purpose statement printed on the cover of every copy, is “Another Testament of Jesus Christ.”
  Each of these three testaments is a part of the great, indivisible whole of the Lord’s revealed word to His children. They contain the words of Christ, which we have been admonished to feast upon as a means of qualifying for eternal life (see 2 Nephi 31:20)." ("The Miracle of the Bible", Ensign, May 2007)

