Elijah and the Baptist
Malachi promises Elijah's coming; the Baptist denies being him while Jesus says that he is (Matthew 11:14; 17:12) — identity, typology, or unresolved tension.
John 1:19-21
19This is John’s testimony, when the Jews sent priests and Levites from Jerusalem to ask him, “Who are you?”
20He declared, and didn’t deny, but he declared, “I am not the Christ.”
21They asked him, “What then? Are you Elijah?” He said, “I am not.” “Are you the prophet?” He answered, “No.”
Malachi 4:5-6
5Behold, I will send you Elijah the prophet before the great and terrible day of Yahweh comes.
6He will turn the hearts of the fathers to the children, and the hearts of the children to their fathers, lest I come and strike the earth with a curse.”
What each tradition says
Rabbinic / Midrashic
The source says Elijah's activity climaxes in his appearance shortly before the Messianic time. Ben Sira says he is appointed to lead the coming ages aright and restore the tribes of Jacob. In the second half of the first Christian century it was expected that Elijah would appear shortly before the coming of the Messiah.
Emil G. Hirsch, Eduard König, Solomon Schechter, Louis Ginzberg, M. Seligsohn, Kaufmann Kohler, The Jewish Encyclopedia (New York: Funk & Wagnalls, 1901-1906), s.v. 'Elijah' — Section 'Elijah as the Forerunner of the Messiah'
Patristic
Augustine explains that John said he was not Elias, while the Lord said he was, because Christ wished to prefigure His own advent in John and to say that John was in the spirit of Elias. What John was to the first advent, Elias will be to the second.
Augustine of Hippo (354-430), Tractates on the Gospel of John, in NPNF ser. 1, vol. 7 (ed. Philip Schaff; New York: Christian Literature Publishing Co., 1886) — Tractate IV on John, §5 (on John 1:19-21)
Catholic
Souvay holds that John denied he was Elias, and Jesus did not admit it, though His words at first seemed to. The disciples understood Jesus spoke of John, meaning Elias was not to come in the flesh; before the multitude, Jesus called John Elias figuratively.
Charles L. Souvay, C.M., The Catholic Encyclopedia, vol. 8 (New York: Robert Appleton Company, 1910), article 'St. John the Baptist'; New Advent edition — Article 'St. John the Baptist', on John 1:19-23 and Matthew 11:14; 17:11-13
Orthodox
Chrysostom says Christ solves the question by distinguishing two comings: Elijah will come before Christ's second advent, and Elijah has now come, referring to John. John is Elijah in this sense, while the Tishbite himself is still to come.
John Chrysostom (c. 349-407), Homilies on the Gospel of Saint Matthew, in NPNF ser. 1, vol. 10 (ed. Philip Schaff; New York: Christian Literature Publishing Co., 1886) — Homily LVII (on Matthew 17:10-13)
Reformed / Protestant
Calvin says John’s reply that he is not Elijah is just and true, because John speaks according to the opinion they attached to the words. But Christ, giving the true interpretation of the Prophet, affirms that John is Elijah.
John Calvin (1509-1564), Commentary on the Gospel according to John, vol. 1 (trans. William Pringle; Edinburgh: Calvin Translation Society) — Commentary on John 1:21
Evangelical
Barnes says the people expected the actual Elijah returned from heaven, and John denied being Elijah in that sense. John did not deny being the Elijah the prophet intended; he corrected their false notion and showed he was the one predicted to prepare the Lord’s way.
Albert Barnes (1798-1870), Notes on the New Testament, Explanatory and Practical (enlarged ed. by Robert Frew; CCEL carries the Baker reprint) — Note on John 1:21