Why do Orthodox Jews not believe in the New Testament?
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Jews do not believe there's anything special about the Christian scriptures because they make claims counter to torah (like that God would abandon the covenant he made with Jews and impose new surprise requirements later). God's torah is true, and any other document that contradicts it therefore doesn't reflect God's truth.
For why one of their most important claims is a non-starter for Jews, see Why don't Jews think Jesus is the messiah?.
There are multiple reasons, far too many to explain in a simple answer. It mostly has to do with various new-testament proofs, that are either not proofs, or were not prophesized to be proofs. There are several important inconsistencies as well that are problematic from a jewish perspective (not just orthodox).
One of those is that mashiach needs to be from David's bloodline (e.g. bereshit 49:10, Yirmiyahu 33:17) through Solomon (e.g. Divrei Hayamim I 22:10. Unfortunately, from the two gospels that show bloodlines, one is through Maria, which is invalid to us because tribal descent is patrilineal (Bamidbar 1:18), and tribal descent can not be adopted (an adopted non-cohen child by a cohen is not given cohen-status) (and even it it could: the line is not through Solomon, but Nathan), and the line through Joseph goes through King Conyahu (Jeconia) that was cursed and kingship was taken away from him (Yirmiyahu 22:30).
There are various of such things: one of which is alleged virgin birth, which is not something than can be observed by onlookers (does a virginborn look any different from a nonvirginborn?) and goes against the principle of public signs from hashem.
Miracles were never part of the necessary conditions for proving someone is mashiach; the Torah even states that G'd will let false prophets perform miracles to test the jewish nation devarim 13:2-6; so we reject the claim that if someone is able to perform miracles, he must be the mashiach.
More importantly: the maschiach was supposed to rebuild the temple (michah 4:1)(the physical one), whereas the temple was still standing in the time of the early gospel; and peace on earth did not happen (michah 4:3,Yeshayahu 2:4); worse: the jewish people saw one of its worse destructions; the romans squashed all rebellion, destroyed our temple, and scattered us amonst the nations. There is not a 'he will come to finish the job' clausule in our texts.
All in all, it is best not to enter a discussion of such things. I personally have high respect for christians and have no interest in convincing them out of their belief. I just belief, as do most jews in my area, that christianity is for the christians, and judaism is for jews; those who were lost to us will find their way back, but generally each nation has its own G'dly-mission on this planet.
Christians believe that Jesus was the Moshiach (Messiah), and that he will return to life at some point in time. However, one of the things that the Moshiach will do is eliminate war. Since Jesus did not do this, he could not have been the Moshiach. Since the New Testament claims that he was the Moshiach, to Orthodox Jews, this invalidates the entire New Testament.
The "Old" testament, whilst containing partly a history of recorded events and also writings of psalms etc. is primarily written by Prophets of G-d. It is part of our "tradition" that prophecy ended with Malachi, who is in the "old" testament. Anything after that is not "divine" and if it claims to contain prophecies, these are always considered false. This even includes the Maccabians, which may be historically accurate and they may even have been religiously observant but is not part of our "canonised" text.
Jesus is not accepted as "Moshiach", as he fails almost all the criteria. (Aside from being Jewish and possibly of the lineage of King David, he has failed every other aspect of what the Moshiach must do).