New Testament Poetry In the Old Testament, there are lengthy poetic passages or entire books of poetry (e.g. Song of Songs). What poetry, if any is there in the New Testament?

John 1:1-18 and Luke 1:46-55 seem like candidates to me.
1 Corinthians chapter 13 is called the Hymn of Love and is thought to be a pre-Pauline hymn that Paul was quoting. Perhaps the most beautiful English rendition of the poem or song is the the King James Bible, but unfortunately it translates ἀγάπην as 'charity', rather that 'love'. The Hymn of Love, from the KJV, amended to speak of 'love':

1 Though I speak with the tongues of men and of angels, and have not love, I am become as sounding brass, or a tinkling cymbal.
2 And though I have the gift of prophecy, and understand all mysteries, and all knowledge; and though I have all faith, so that I could remove mountains, and have not love, I am nothing.
3 And though I bestow all my goods to feed the poor, and though I give my body to be burned, and have not love, it profiteth me nothing.
4 Love suffereth long, and is kind; love envieth not; love vaunteth not itself, is not puffed up,
5 Doth not behave itself unseemly, seeketh not her own, is not easily provoked, thinketh no evil;
6 Rejoiceth not in iniquity, but rejoiceth in the truth;
7 Beareth all things, believeth all things, hopeth all things, endureth all things.
8 Love never faileth: but whether there be prophecies, they shall fail; whether there be tongues, they shall cease; whether there be knowledge, it shall vanish away.
9 For we know in part, and we prophesy in part.
10 But when that which is perfect is come, then that which is in part shall be done away.
11 When I was a child, I spake as a child, I understood as a child, I thought as a child: but when I became a man, I put away childish things.
12 For now we see through a glass, darkly; but then face to face: now I know in part; but then shall I know even as also I am known.
13 And now abideth faith, hope, love, these three; but the greatest of these is love.

1 Peter 3:18-22, which Burton L. Mack, in Who Wrote the New Testament, page 91, calls a ‘Christ hymn’, and says is a genre that became popular among early Christians.

18 For Christ also hath once suffered for sins, the just for the unjust, that he might bring us to God, being put to death in the flesh, but quickened by the Spirit:
19 By which also he went and preached unto the spirits in prison;
20 Which sometime were disobedient, when once the longsuffering of God waited in the days of Noah, while the ark was a preparing, wherein few, that is, eight souls were saved by water.
21 The like figure whereunto even baptism doth also now save us (not the putting away of the filth of the flesh, but the answer of a good conscience toward God,) by the resurrection of Jesus Christ:
22 Who is gone into heaven, and is on the right hand of God; angels and authorities and powers being made subject unto him.

A form of ancient poetry, quite different to modern ideas of poetry was the chiasm (from the Greek letter chi, χ), or chiastic structure. In these structures, an opening set of events is mirrored by a second set. They were used extensively in the Old Testament and, to a lesser extent, in the New Testament, but perhaps no more so than in Mark's Gospel. Here is an example paraphrased from Mark 3:20-4:1 (with three verses of prose interposed at 3:27-29):

a  The multitude is so thick that they could not so much as eat bread (3:20)
.b Jesus' friends say he is beside himself (3:21)
..c    Scribes say Jesus is possessed and by the devil casts out devils (3:22)
...d   How can Satan cast out Satan? (3:23)
....e  If a kingdom be divided the kingdom can not stand (3:24)
....e' If a house be divided the house can not stand (3:25)
...d'  If Satan is divided he can not win (3:26)
..c'   Because they said Jesus has an unclean spirit (3:30)
.b'    Jesus' family comes to him. Jesus says his followers are his family (3:31-35)
a' The multitude is so thick that Jesus enters a boat (4:1)

Luke's Magnificat is a poem modelled on Hannah's prayer from the Old Testament. It is
the first of four canticles in the opening of Luke’s Gospel and is a beautiful hymn spoken by Mary, at Luke 1:46-55:-

46 And Mary said, My soul doth magnify the Lord,
47 And my spirit hath rejoiced in God my Saviour.
48 For he hath regarded the low estate of his handmaiden: for, behold, from henceforth all generations shall call me blessed.
49 For he that is mighty hath done to me great things; and holy is his name.
50 And his mercy is on them that fear him for all generations.
51 He hath shewed strength with his arm; he hath scattered the proud in the imagination of their hearts.
52 He hath put down the mighty from their seats, and exalted them of low degree.
53 He hath filled the hungry with good things; and the rich he hath sent empty away.
54 He hath holpen his servant Israel, in remembrance of his mercy;
55 As he spake to our fathers, to Abraham, and to his seed for ever.