What is the Biblical basis for saying that God can annihilate human souls?

The idea of eternal conscious torment is one of those traditions carried over from medieval Catholicism (ultimately coming from Greek philosophy) that's heavily defended in Protestantism but not really taught in Scripture, or if it is, not plainly, as it is only inferred from certain expressions like "unquenchable fire" which speaks of the fire being unquenchable but does not say it will not burn anyone up. It can be defended with proof-texts to be sure, but is it explicitly and clearly taught? No. The annihilation of the souls of the wicked, however, is explicitly taught in at least two verses: 0 vs 2.

In large part the idea of eternal conscious torment in hell is based on the Platonic concept of the natural immortality of the soul. Plato and the Platonizing "church fathers" believed the soul to be so immortal by its very nature that God could not destroy it even if he wanted to. The result is that all God can do with the souls of the wicked in this philosophy is to dump them somewhere for all eternity. (In Platonism the soul was conceived as actually being a breakaway from God's substance or essence, thus sharing the nature of God and indestructible.)

But does scripture teach the natural immortality of the soul, or does it teach that souls are only immortal so long as God deems them to be so by his will?

1 Timothy 6:15-16 NIV "which God will bring about in his own time—God, the blessed and only Ruler, the King of kings and Lord of lords, who alone is immortal and who lives in unapproachable light, whom no one has seen or can see. To him be honor and might forever. Amen."

According to Scripture, God alone is immortal by nature. The soul then is only conditionally immortal. That is, only immortal if God says so. That means the biggest presupposition that caused the "church fathers" to develop the theory of eternal conscious torment is false, according to scripture.

Once you understand that, you will find a different story in Jesus' sayings on the subject. You will be free from the tyranny of Platonic philosophy and tradition based thereon, which before prevented you from taking Jesus at his word when he said:

Matthew 10:28 NIV "Do not be afraid of those who kill the body but cannot kill the soul. Rather, be afraid of the One who can destroy both soul and body in hell."

He said "destroy" but tradition has always forced us to interpret "destroy" as "torment"! If we re-evaluate the tradition, and correct the root problem (Platonic misunderstanding of the nature of the soul) we no longer have to force ourselves to misinterpret the Bible in this way.

There's an entire website dedicated to a growing movement of Protestants who are Rethinking Hell, which is the name of the site: http://rethinkinghell.com/

And furthermore, there is one verse in the Bible that positively clinches the case by explicitly stating the wicked will be devoured by fire:

But the wicked shall perish, and the enemies of the Lord shall be as the fat of lambs: they shall consume; into smoke shall they consume away. (Psalm 37:20 KJV)

But the wicked will perish: Though the Lord’s enemies are like the flowers of the field, they will be consumed, they will go up in smoke. (Psalm 37:20 NIV)

In your other question about hell, Would Jesus weep for the souls in Hell? I noticed the following. You quote a girl who claims to have visited hell as saying that Jesus said to her:

because of sin and the lack of repentance, humanity ends here, and there are more that perish than those who reach My glory

Notice how even in the traditional Platonic paradigm the scriptural word "perish" (which means to cease to exist) is used, but what they're talking about isn't perishing, its continuing to be tormented forever and ever. The fact that "perish" is used is a clear demonstration that eternal conscious torment is not the original doctrine. If it were the original doctrine then "torment" would be the word used in scripture and would have become the word traditionally used to describe the fate of the lost, but since "perish" persists as the word of choice, being the scriptural word, we know that the present traditional doctrine arose after the New Testament was written, and after this language of "perish" was already so well established that it could not be replaced. The same argument applies to the word "destroy" in Matthew 10:28.

To add to Gordon James' well-sourced answer, I'd like to address a few of the remarks expressed in the question directly.

"Why can't God simply make us all vanish? Maybe because God is not able to!"

God certainly can. "Do not fear those who kill the body but cannot kill the soul. But rather fear Him who is able to destroy both soul and body in hell." Matthew 10:28

"If God were capable of annihilating the soul, He would have done it long ago."

At the cross, Jesus answered the question once and for all who God was. He proved that God was love (through self-sacrifice) and God was righteousness (by keeping His laws perfectly and living a life of righteousness).

Had God destroyed mankind before this, He would not have had the chance to demonstrate to intelligent creations neither character to the same extent.

It can be questioned whether even angels truly understood before the plan of salvation was revealed, since a third of them fell, unable to withstand Satan's sophistry (Rev 12:4). Paul says of the gospel, "Even angels long to look into these things." (1 Peter 1:12 NIV)

Therefore, when Satan was finally defeated at the cross, God's character was fully revealed. Yet, God still prolongs the day of judgement to this day... why? purely out of mercy...

"The Lord is not slow in keeping his promise, as some understand slowness. Instead he is patient with you, not wanting anyone to perish, but everyone to come to repentance." (2 Peter 3:9)

First, the idea of the immortality of the soul comes from the Greek dualistic view of body and soul as separate, with a mortal body and an immortal soul.

Immortality belongs to God alone (I Tim. 6:16) in the sense of before creation and on to eternity.

God offers eternal life (John 3:15–16; 10:28; 17:2; Rom. 2:7; 6:23; 1 Cor. 15:42f; 50, 54; Gal. 6:8; 1 John 5:11).

Many will not be saved - look at Mt 7:13-14 “Enter through the narrow gate; for the gate is wide and the way is broad that leads to destruction, and there are many who enter through it. For the gate is small and the way is narrow that leads to life, and there are few who find it.

The "great white throne judgment" in Rev 20:11-15 tells about the second death.

11 Then I saw a great white throne and Him who sat upon it, from whose presence earth and heaven fled away, and no place was found for them. 12 And I saw the dead, the great and the small, standing before the throne, and books were opened; and another book was opened, which is the book of life; and the dead were judged from the things which were written in the books, according to their deeds. 13 And the sea gave up the dead which were in it, and death and Hades gave up the dead which were in them; and they were judged, every one of them according to their deeds. 14 Then death and Hades were thrown into the lake of fire. This is the second death, the lake of fire. 15 And if anyone’s name was not found written in the book of life, he was thrown into the lake of fire.

Calling this the second death is a strong indication that the unsaved do not have an eternal existence of eternal torture.

The Old Testament speaks of destruction as the "reward for wickedness" (look here for a more complete treatment).

The New Testament gives many pictures of destruction. Look at James 4:12

12 There is only one Lawgiver and Judge, the One who is able to save and to destroy; but who are you who judge your neighbor?

Here it seems like salvation or destruction are the only options.

Here are a few scriptures that speak of destruction

Psalm 1:6 ... but the way of the ungodly shall perish

Psalm 37:20 But the wicked shall perish... they shall consume; into smoke shall they consume away.

Psalm 92:7 ... shall be destroyed forever

Matthew 10:28b Rather, fear him which is able to destroy both soul and body in hell.

John 3:16 ... whosoever believeth in him should not perish (Greek: destroyed)

Romans 6:23 For the wages of sin is death …

Philippians 3:19 whose end is "destruction"

2 Thessalonians 1:9 who shall be punished with everlasting destruction ...

Hebrews 10:39 But we are not of them who draw back unto perdition (Greek: destruction); but of them that believe to the saving of the soul.

James 4:12a There is one lawgiver, who is able to save and to destroy.

Revelation 20:14 This is the second death...

Greg Boyd has put together a list of scriptures here.

A good book on the subject is E.W. Fudge & R.A. Peterson, "Two views of Hell: A Biblical and Theological Dialog," Inter Varsity Press, (2000)

In this book we read . . .

"The fact is that the Bible does not teach the traditional view of final punishment. Scripture nowhere suggests that God is an eternal torturer. It never says the damned will writhe in ceaseless torment or that the glories of heaven will forever be blighted by the screams from hell. The idea of conscious everlasting torment was a grievous mistake, a horrible error, a gross slander against the heavenly Father, whose character we truly see in the life of Jesus of Nazareth." (p. 21)