The New Testament's own use of Elijah is as a forerunner, and that reading is not imposed on the text from outside. Malachi closes the Old Testament by promising that God will send Elijah before the great and awesome Day of the LORD. Gabriel tells Zechariah that his son will go before the Lord "in the spirit and power of Elijah" and quotes Malachi's own words back to him (Luke 1:17). Jesus says of John, "he is the Elijah who was to come" (Matthew 11:14), and later, "Elijah has already come, and they did not recognize him" (Matthew 17:12), and Matthew records that the disciples understood He was speaking about John.
Scripture puts a second half beside that one, and both halves stand. Asked directly whether he was Elijah, John answered "I am not" (John 1:21). From its earliest centuries the church read the identification as one of office and spirit rather than of person, and said so against the idea that a soul had come back into another body. Gregory the Great settled it in one sentence in the sixth century: John was Elijah in spirit and he was not Elijah in person, so what the Lord declares about the spirit, John denies about the person. Origen had already drawn the distinction Luke 1:17 itself draws, that the angel said spirit and power and did not say mind. The resemblance a reader can check for himself is in the clothing. 2 Kings 1:8 describes a hairy man with a leather belt around his waist, and Matthew 3:4 describes John in camel's hair with a leather belt around his waist. Bede listed the parallels in the eighth century and added the one that lands hardest: both men were hunted by a king and his queen.
On the mountain of the transfiguration, Moses and Elijah stand with Jesus, and Luke alone reports what they were talking about, "His departure, which He was about to accomplish at Jerusalem" (Luke 9:31). Malachi's last chapter had named the same two men in the same order, the law of Moses at 4:4 and Elijah at 4:5. John Chrysostom's fourth-century answer to why these two and no others is concrete: both had stood up to a king on behalf of an ungrateful people, both were plain men, and neither owned anything, Moses possessing nothing and Elijah nothing but his sheepskin. Augustine of Hippo read the voice from the cloud as the point of the whole scene, noticing that it did not say these are my beloved sons, because only one is the Son and the others are adopted. The passage gives its own last word at Matthew 17:8, "And when they looked up, they saw no one except Jesus." Which mountain it was is not stated in Scripture. Tabor carries a fourth-century tradition and Hermon is argued from its nearness to Caesarea Philippi, and this page names both and settles neither.
Elijah went up and did not die, and the church has long read that as a figure of the ascension of Christ. John Wesley wrote that by translating him God gave, in a dark and degenerate age, a very sensible proof of another life, together with a type of the ascension of Christ and the opening of the kingdom of heaven to all believers. Albert Barnes is blunt about the ground the figure stands on: "No honest exegesis can explain this passage in any other sense than as teaching the translation of Elijah, who was taken from the earth, like Enoch Gen 5:24, without dying." The difference matters and this page states it plainly. Elijah was taken. Christ ascended in His own right, having died and risen first.
Two of Elijah's miracles are picked up in the Gospels. Jesus took the widow of Zarephath into the synagogue at Nazareth and said that Elijah was sent past every widow in Israel to a Sidonian one, and the congregation tried to throw Him off a cliff for saying it (Luke 4:25-29). His point was that God's mercy in Elijah's day reached past Israel's borders. It was never that Israel had been set aside, and the next place the New Testament quotes Elijah says so in Paul's own words. At the gate of Nain, Luke has Jesus meet the funeral of a widow's only son and closes the account with the same clause 1 Kings 17:23 closes on, that He gave him back to his mother. The echo is deliberate and so is the difference. Elijah stretched himself over the child three times and pleaded; Jesus touched the coffin and spoke.
Paul takes Elijah's worst moment and makes it the proof that God has not rejected His people. Romans 11 quotes the complaint at Horeb, then quotes God's answer, the seven thousand, and calls it the pattern for a remnant chosen by grace in Paul's own day. Theodoret of Cyrus drew the edge of it in the fifth century, that Elijah did not know about those seven thousand until God told him, so a reader today should not assume he knows the count either. James takes the other end of the life. He names Elijah as a man just like us, points at the drought and the rain, and attaches the example to an instruction in the present tense: confess your sins to each other, pray for each other, and the prayer of a righteous man has great power to prevail. The argument runs from Elijah's ordinariness to the reader's own praying, and this site teaches it at that strength. Bede reached the same conclusion in the eighth century and pressed it further, that if one man praying obtained so much, the frequent prayer of many righteous people is worth more.