Regarding the New Jerusalem, Revelation 21:25 states:
Its gates will never be closed during the day (and there will be no night there). (Revelation 21:25 NET)
What is the significance of the gates never being closed?
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The main reason to close the gates of a city would be to protect those living within from harm (such as from invaders). The gates of the New Jerusalem will never be closed since there will never be any fear of war or invasions. The gates always remaining open signify perpetual peace.
Also note that even in times of peace, city gates would still be closed at night. But "there will be no night there" gives another reason why the New Jerusalem's gates will never be closed, for "the Lamb is its light" (NKJV, end of verse 23).
An excerpt from Albert Barnes commentary on verse 25:
It shall be all day; all unclouded splendor. When, therefore, it is said that the gates should not be “shut by day,” it means that they would never be shut. When it is said that there would be no night there, it is, undoubtedly, to be taken as meaning that there would be no literal darkness, and nothing of which night is the emblem: no calamity, no sorrow, no bereavement, no darkened windows on account of the loss of friends and kindred.
A walled city with closed gates would be a safe haven, a place of refuge from enemies and wild animals. If someone lived outside the city in a tent or other temporary shelter, the walled city would also offer more protection from the natural elements. As Brian's answer notes, gates that never close signify perpetual peace: no more attacks from the enemy, no more wild animals, no more storms. There is nothing that would cause a person living outside the city to seek refuge.
There is more.
Closed gates also prevented good people who wanted to come into the city from entering. If someone arrived after the gates closed, they were denied protection the city offered. Also closed gates prevented people from leaving.
Therefore gates that never shut also signify perfect freedom and unrestricted access to the city and all that is there.
For many years I didn't bother trying to tackle Revelation because it seemed to lack the little arrow that you see on maps that say "You are here". Rather than it being a "revelation" it appeared to be an impenetrably obscure riddle, until one day a "You are here" arrow appeared while I was reading 1 Corinthians:
1Co 15:24 Then the end will come, when after he has done away with every ruler and every authority and power, the Messiah hands over the kingdom to God the Father. 1Co 15:25 For he must rule until God puts all the Messiah's enemies under his feet. 1Co 15:26 The last enemy to be done away with is death, 1Co 15:27 for "God has put everything under his feet." Now when he says, "Everything has been put under him," this clearly excludes the one who put everything under him. 1Co 15:28 But when everything has been put under him, then the Son himself will also become subject to the one who put everything under him, so that God may be all in all.
This passage was and is to me the key to understanding the second section (after the letters to the seven Asian assemblies). The pouring out of the bowls, etc. is the prophetic account of God using broad-brush terrorism from his sky kingdom to demoralize the nations with violence and disasters to render even the most powerful and stubborn rebellious enemies of God to become reluctantly obedient to God's Christ:
Psa_2:9 Thou shalt break them with a rod of iron; thou shalt dash them in pieces like a potter's vessel. ... Rev_2:27 And he shall rule them with a rod of iron; as the vessels of a potter shall they be broken to shivers: even as I received of my Father.
Rev_12:5 And she brought forth a man child, who was to rule all nations with a rod of iron: and her child was caught up unto God, and to his throne.
Rev_19:15 And out of his mouth goeth a sharp sword, that with it he should smite the nations: and he shall rule them with a rod of iron: and he treadeth the winepress of the fierceness and wrath of Almighty God.
This is the same strategy that Bin Laden, Daesh and every terrorist uses: intimidation through horrific designs. God is the ultimate terrorist:
Jos 10:19 but don't stay there yourselves. Instead, pursue your enemies and attack them from behind. Don't allow them to enter their cities, because the LORD your God has delivered them into your control." Jos 10:20 Now it came about that after Joshua and the Israelis had finished the battle, destroying and scattering their survivors, who retreated into their fortified cities, Jos 10:21 the entire army returned safely to Joshua's encampment at Makkedah. No one could speak so much as a single word against any of the Israelis. Jos 10:22 Then Joshua gave this order: "Unseal the mouth of the cave and bring out these five kings to me from the cave." Jos 10:23 So they did. They brought out these five kings to him from within the cave: the king of Jerusalem, the king of Hebron, the king of Jarmuth, the king of Lachish, and the king of Eglon. Jos 10:24 When they had brought these kings out to Joshua, Joshua called for all the men of Israel and spoke to the leaders of the men who had gone out to war along with him, "Come close and put your feet on the necks of these kings." So they came near and put their feet on their necks. Jos 10:25 Joshua told the army, "Don't fear or be dismayed! Be strong and courageous, because this is how the LORD will treat all of your enemies whom you fight." Jos 10:26 After this, Joshua struck those kings down, executing them, and hanged them on five gallows until sunset.
So after the demoralization of the nations, when God has subjugated every haughty royal, national, corporate and religious person and institution and reduced them to submissive putty under the heel and iron rod of his Christ then God himself descends to the middle east and establishes his everlasting home amidst his holy people. His home in the heavenly Jerusalem descends, Jesus goes back to his brethren, the saints and all God's people live in bliss in the many rooms in Jesus' father's house forever.
But outside of the holy city are the "dogs" - the broken rebels. They may not enter the city, though they must appear to bring Yehovah tribute:
Rev 21:24 The nations will walk in its light, and the kings of the earth will bring their glory into it. Rev 21:25 Its gates will never be shut at the end of the day—because there will be no night there. Rev 21:26 People will bring the glory and wealth of the nations into it. Rev 21:27 Nothing unclean, or anyone who does anything detestable, and no one who tells lies will ever enter it. Only those whose names are written in the lamb's Book of Life will enter it.
So the open gate is a symbol of the complete and utter impotence of the once fierce enemies of God and the eternal "rest" for the people of God. Joshua putting his feet on the necks of his enemies presages God putting all things under the feet of his Christ:
Heb 4:8 For if Joshua had given them rest, he would not have spoken later about another day. Heb 4:9 There remains, therefore, a Sabbath rest for the people of God, Heb 4:10 because the one who enters God's rest has himself rested from his own actions, just as God did from his.
The open gate is more than a symbol. The gate is open because God's enemies will never rise again while God's people enjoy God's promised everlasting rest in the promised land:
1Ch 23:25 For David had said "The LORD God of Israel has granted rest to his people, and he has taken Israel as his eternal residence.
Isa 62:1 "And I won't remain silent, and for Jerusalem's sake I won't stay quiet, until her vindication shines out like brightness, and her salvation like a burning torch. Isa 62:2 The nations will see your vindication, and all the kings your glory; and people will call you by a new name that the mouth of the LORD will bestow. Isa 62:3 You will be a crown of splendor in the LORD's hand, and a royal diadem in the hand of your God. Isa 62:4 And you'll no longer be called 'Deserted,' and your land will no longer be called 'Desolate'; but people will call you 'Hephzibah,' and your land 'Beulah'—for the LORD will take delight in you, and your land will be married." Isa 62:5 "For just as a young man marries a maiden, so your sons will marry you; and just as a bridegroom rejoices over his bride, so your God rejoice will over you. Isa 62:6 "Upon your walls, Jerusalem, I have posted watchmen; all day and all night they won't remain silent. You who make mention of the LORD, take no rest, Isa 62:7 and give him no rest until he prepares, establishes and makes Jerusalem a song of praise throughout the earth. Isa 62:8 "The LORD has sworn by his right hand and by his mighty arm: 'I will never again give your grain as food for your enemies; never again will foreigners drink your new wine for which you have toiled; Isa 62:9 but surely those who harvest it will eat it and praise the name of the LORD, and those who gather it will drink it in the courts of my sanctuary,' says your God." Isa 62:10 "Pass through the gates! prepare the way for the people! Build up! Build up the highway! Clear it of stumbling stones, speak among the peoples. Isa 62:11 Here is the LORD! Proclaim to the ends of the earth, say to the inhabitants of Zion: 'See, your salvation is coming! See, his reward is with him, and his recompenses are before him.' Isa 62:12 People will call them, 'The Holy People,' 'The Redeemed of the LORD'; and they will call you, 'Sought After,' 'The City Not Deserted.'"