This is rather easy, as the belief in reincarnation can only be (erroneously) held if no attentive and systematic reading of Scriptures has taken place. What generally happens is that a “hearsay Christian” (vast majority nowadays, I’m afraid) reads or receives from some friend some information about verses of the Gospel in which Jesus would seem to endorse the theory of reincarnation. Also rather spread are legends about the bible having been “manipulated” (an old dish of heresy, this one), but I am going to discard those for their evident absurdity.
One verse often cited is the one in which Jesus asks the Apostles who they think He is and they answer ” Some say John the Baptist, others Elijah, still others Jeremiah or one of the prophets.” (Mt. 16, 13-14). To an attentive reader it is evident here that Jesus could not possibly have been the reincarnation of John the Baptist, who lived in the same time as Jesus. Therefore, in order not to make the entire construct absurd the phrase must be read as “some believe you have in you the spiritual strength and power of John The Baptist (who had been executed already), some of Elijah, etc.).” This reading makes much more sense than reincarnation, which is in light of the pure historical event of St. John’s life utterly absurd. Also note that Jesus here does not say that what people believe is right. He says instead, emphatically so, that what Peter says is right. Peter doesn’t say anything at all about reincarnation. What he says is: “Thou art the Christ, the Son of the living God!”. There can be no more powerful evidence that Jesus, by so emphatically endorsing Peter’s claim, utterly disregards all the claims formerly made.
Another rather difficult part is John 9: 2 where Jesus is asked: “Rabbi, who sinned, this man or his parents, that he was born blind?” If the man was born blind, goes the reasoning, then he can only have sinned in another life, which would prove at least that belief in reincarnation was spread in Jesus’ time. By answering “neither this man nor his parents sinned”, Jesus would thus implicitly accept the theory as he doesn’t explicitly correct it. Against such interpretation can be said: 1) that some Jews believed that a baby could sin in the womb, a theory known to Jesus and all present. 2) that there is no evidence in jewish sources that a belief in reincarnation was spread; 3) that Jesus does not endorse everything which he does not explicitly refuse, as we have seen in the former passage in which he doesn’t waste time in explicitly saying that he is not ” a new Elijah”, etc. When Jesus wants to teach, he always takes care to make the point.
Which last point takes us neatly to, well, the point. It is utterly illogical to give an adventurous interpretation to one or two Gospel passages, by at he same time disregarding the entire New Testament. Jesus alone talks of Hell, eternal punishment, Gehenna, fire everlasting & Co. more than any other in the Bible. The examples are too numerous and too well-known to limit ourselves to individual examples. If there is one leitmotiv in the Gospel, it is atonement, redemption, hell and heaven. It is immediately obvious on serious reflection that Christ’s death on the Cross doesn’t make any sense if reincarnation operates anyway; nor does His continuous, insisted mentioning of hell and eternal punishment. Here we see another sign of the times: that Jesus spoke so often of hell is nowadays completely disregarded; it has just disappeared from the radar screen to make place for platitudes entirely devoid of context a’ la “do not judge”, the rhetoric of peaaace and the like.
The concept of atonement is so elementary to a practising Catholic that it would barely be worth the mentioning. But in non-practising Catholics, and in non-Catholic Christians, the concept of atonement and redemption opening the way to everlasting life in God can be extremely diluted or even forgotten, with Jesus reduced to the task of spreading some rather good news and telling everyone to behave and be “inclusive” and “tolerant”. Just ask every non churchgoer to tell you in few words why Jesus came to Earth and stun in disbelief at the answers you hear.
Further difficulties for such interpretative “adventures” arise if we read further in the New Testament. “It is appointed that human beings die once, and after this the judgment”, says in Hebrew 9, 27. The author of this (be he St. Paul or not) is one of the first Christians. He lived in close contact with many of the (other) Apostles. This is real, first hand evidence of what the First Christians were ready to die for. Would Jesus ever accept reincarnation “by implication” but allow His disciples to be, in such an important matter, so tragically and completely wrong? Thought not…
In short, the belief in reincarnation is totally unscriptural not only because Jesus never endorsed it; not only because Jesus himself was extremely clear in his warning about eternal punishment; not only because the belief in eternal damnation was obviously commonly accepted among his disciples, but because if one accepts the theory, the entire mission and sacrifice of Jesus loses its meaning.