Admittedly I am an animal lover and have had beloved pets all my life, however, I personally believe that none of the current animals that we have shall exist in the eternity.
The most common scriptures to use to prove animals shall be is in Isaiah:
Isa 11:6-10
6 The wolf also shall dwell with the lamb, and the leopard shall lie down with the kid; and the calf and the young lion and the fatling together; and a little child shall lead them.
7 And the cow and the bear shall feed; their young ones shall lie down together: and the lion shall eat straw like the ox.
8 And the sucking child shall play on the hole of the asp, and the weaned child shall put his hand on the cockatrice' den.
9 They shall not hurt nor destroy in all my holy mountain: for the earth shall be full of the knowledge of the LORD, as the waters cover the sea.
10 And in that day there shall be a root of Jesse, which shall stand for an ensign of the people; to it shall the Gentiles seek: and his rest shall be glorious. (KJV)
I personally believe these are metaphors and not literal examples. I think we all know that no children will be born or exist at that time so I believe we can assume the point here is that there will be complete peace and safety, explained in a way that people of that day could understand.
Here are some verses that I believe show differently:
Jeremiah 7:20 Therefore thus saith the Lord GOD; Behold, mine anger and my fury shall be poured out upon this place, upon man, and upon beast, and upon the trees of the field, and upon the fruit of the ground; and it shall burn, and shall not be quenched.
Zephaniah 1:1 The word of the LORD which came unto Zephaniah the son of Cushi, the son of Gedaliah, the son of Amariah, the son of Hizkiah, in the days of Josiah the son of Amon, king of Judah.
Zephaniah 1:2 I will utterly consume all things from off the land, saith the LORD.
Zephaniah 1:3 I will consume man and beast; I will consume the fowls of the heaven, and the fishes of the sea, and the stumblingblocks with the wicked; and I will cut off man from off the land, saith the LORD.
Zephaniah 1:4 I will also stretch out mine hand upon Judah, and upon all the inhabitants of Jerusalem; and I will cut off the remnant of Baal from this place, and the name of the Chemarims with the priests;
Zephaniah 1:5 And them that worship the host of heaven upon the housetops; and them that worship and that swear by the LORD, and that swear by Malcham;
Zephaniah 1:6 And them that are turned back from the LORD; and those that have not sought the LORD, nor enquired for him.
Zephaniah 1:7 Hold thy peace at the presence of the Lord GOD: for the day of the LORD is at hand: for the LORD hath prepared a sacrifice, he hath bid his guests.
Zephaniah 1:8 And it shall come to pass in the day of the LORD'S sacrifice, that I will punish the princes, and the king's children, and all such as are clothed with strange apparel.
Zephaniah 1:9 In the same day also will I punish all those that leap on the threshold, which fill their masters' houses with violence and deceit.
Zephaniah 1:10 And it shall come to pass in that day, saith the LORD, that there shall be the noise of a cry from the fish gate, and an howling from the second, and a great crashing from the hills.
Zephaniah 1:11 Howl, ye inhabitants of Maktesh, for all the merchant people are cut down; all they that bear silver are cut off.
Zephaniah 1:12 And it shall come to pass at that time, that I will search Jerusalem with candles, and punish the men that are settled on their lees: that say in their heart, The LORD will not do good, neither will he do evil.
Zephaniah 1:13 Therefore their goods shall become a booty, and their houses a desolation: they shall also build houses, but not inhabit them; and they shall plant vineyards, but not drink the wine thereof.
Zephaniah 1:14 The great day of the LORD is near, it is near, and hasteth greatly, even the voice of the day of the LORD: the mighty man shall cry there bitterly.
Zephaniah 1:15 That day is a day of wrath, a day of trouble and distress, a day of wasteness and desolation, a day of darkness and gloominess, a day of clouds and thick darkness,
Zephaniah 1:16 A day of the trumpet and alarm against the fenced cities, and against the high towers.
Zephaniah 1:17 And I will bring distress upon men, that they shall walk like blind men, because they have sinned against the LORD: and their blood shall be poured out as dust, and their flesh as the dung.
Zephaniah 1:18 Neither their silver nor their gold shall be able to deliver them in the day of the LORD'S wrath; but the whole land shall be devoured by the fire of his jealousy: for he shall make even a speedy riddance of all them that dwell in the land.
Ezekiel 32:13 I will destroy also all the beasts thereof from beside the great waters; neither shall the foot of man trouble them any more, nor the hoofs of beasts trouble them.
Hos 4:3 Therefore shall the land mourn, and every one that dwelleth therein shall languish, with the beasts of the field, and with the fowls of heaven; yea, the fishes of the sea also shall be taken away.
Looks to me that God says he will kill all animals during the 7th trump.
Animals at least for Adam were created as helpmeets, yet they were not enough so Eve was created. I suspect that we will not need animal helpmeets in the eternity.
Ps 49:20
20 Man that is in honour, and understandeth not, is like the beasts that perish.
KJV
Ps 49:12
12 Nevertheless man being in honour abideth not: he is like the beasts that perish.
KJV
Jer 50:3
3 For out of the north there cometh up a nation against her, which shall make her land desolate, and none shall dwell therein: they shall remove, they shall depart, both man and beast.
KJV
Jer 51:62
62 Then shalt thou say, O LORD, thou hast spoken against this place, to cut it off, that none shall remain in it, neither man nor beast, but that it shall be desolate for ever.
KJV
Man as in human beings shall no longer exist either. Human flesh will be no more.
Jer 33:10
10 Thus saith the LORD; Again there shall be heard in this place, which ye say shall be desolate without man and without beast, even in the cities of Judah, and in the streets of Jerusalem, that are desolate, without man, and without inhabitant, and without beast,
KJV
Eccl 3:19-21
19 For that which befalleth the sons of men befalleth beasts; even one thing befalleth them: as the one dieth, so dieth the other; yea, they have all one breath; so that a man hath no preeminence above a beast: for all is vanity.
20 All go unto one place; all are of the dust, and all turn to dust again.
21 Who knoweth the spirit of man that goeth upward, and the spirit of the beast that goeth downward to the earth?
KJV
Man's spirit goes up to heaven, but the spirit of animals go downward. It's God's way. I trust that God knows what's best.
The only mention of an animal in the eternity chapters of Revelation is the Lamb, which is Jesus Christ and not a literal lamb.