Luke 19:36-40: "36 And as he rode along, they spread their cloaks on the road. 37 As he was drawing near - already on the way down the Mount of Olives - the whole multitude of his disciples began to rejoice and praise God with a loud voice for all the mighty works that they had seen, 38 saying, 'Blessed is the King who comes in the name of the Lord! Peace in heaven and glory in the highest!" 39 And some of the Pharisees in the crowd said to him, 'Teacher, rebuke your disciples.' 40 He answered, 'I tell you, if these were silent, the very stones would cry out.'"
John 19:14-19: "14 Now it was the day of Preparation of the Passover. It was about the sixth hour. He [Pilate] said to the Jews, 'Behold your King!' 15 They cried out, 'Away with him, away with him, crucify him!' Pilate said to them, 'Shall I crucify your King?' The chief priests answered, 'We have no king but Caesar.' 16 So he delivered him over to them to be crucified.
"So they took Jesus, 17 and he went out, bearing his own cross, to the place called The Place of a Skull, which in Aramaic is called Golgotha. 18 There they crucified him, and with him two others, one on either side, and Jesus between them. 19 Pilate also wrote an inscription and put it on the cross. It read, 'Jesus of Nazareth, the King of the Jews.'"
Philippians 2:5-8: "5 Have this mind among yourselves, which is yours in Christ Jesus, 6 who, though he was in the form of God, did not count equality with God a thing to be grasped, 7 but made himself nothing, taking the form of a servant, being born in the likeness of men. 8 And being found in human form, he humbled himself by becoming obedient to the point of death, even death on a cross."
Revelation 19:11-16: "11 Then I saw heaven opened, and behold, a white horse! The one sitting on it is called Faithful and True, and in righteousness he judges and makes war. 12 His eyes are like a flame of fire, and on his head are many diadems, and he has a name written that no one knows but himself. 13 He is clothed in a robe dipped in blood, and the name by which he is called is The Word of God. 14 And the armies of heaven, arrayed in fine linen, white and pure, were following him on white horses. 15 From his mouth comes a sharp sword with which to strike down the nations, and he will rule them with a rod of iron. He will tread the winepress of the fury of the wrath of God the Almighty. 16 On his robe and on his thigh he has a name written, King of kings and Lord of lords."
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The text from Luke appears in the first two verses, although the majority of the first verse lists typical parade elements that aren't present at Jesus' entry into Jerusalem: "No tramp of soldiers' marching feet... No sound of music's martial beat... No bells in triumph ring." Although negated, "No city gates swing open wide" from the end of the first verse might refer to Psalm 24:7: "Lift up your heads, O gates! And be lifted up, O ancient doors, that the King of glory may come in."
The second verse mentions some parts of the triumphal entry that aren't in the text cited from Luke: cheering children, palm branches, and Jesus' riding on a colt. The cheering children are mentioned in Matthew 21:15: "But when the chief priests and the scribes saw the wonderful things that he did, and the children crying out in the temple, 'Hosanna to the Son of David!' they were indignant." The Luke account of the triumphal entry is the only one of the four Gospel accounts (Matthew 21:1-11, Mark 11:1-10, Luke 19:28-40, and John 12:12-15) that doesn't mention people's spreading branches. Jesus' riding on a colt is in Luke, but it's in verses 29-35, immediately before the section that's cited.
The text from John appears at the end of the third verse, specifically "The streets with tumult ring, / As Pilate to the mob replies, / 'Behold, behold your King!'" The first half of the verse mentions the crown of thorns and the soldiers' mocking Jesus. This appears in Matthew 27:28-31, Mark 15:16-20, Luke 22:63-65 (although this doesn't mention the crown of thorns), and John 19:2-3.
The text from Philippians (specifically verses 7 and 8) is at the beginning of the hymn's fourth verse: "Now He who bore for mortals' sake / The cross and all its pains / And chose a servant's form to take."
I can't see any textual resemblance between the passage from Revelation and the hymn. My best guess is that it's included as an image of "The King of glory reigns" in the hymn's fourth verse.
