What is the scriptural basis for the conditionalist doctrine of hell? By conditionalist doctrine, I mean the belief that hell is the destruction of those souls that go there, rather than the everlasting torment or eventual purification of said souls.
Technically, a couple of different terms are being confused in the question.

Conditionalism is the teaching that immortality / eternal life is conditional upon a right standing with God.

Annihilationism is the teaching that anyone who is ultimately unrepentant will cease to exist, usually after a period of torment that accommodates to the amount of evil they have done.

While these are technically separate, they do often go hand-in-hand.



Conditionalism

Most defenses for conditionalism start out with a general summary of their main point: nothing in the bible explicitly says humans are inherently immortal; not their bodies, not their spirits, not their souls.

Specific appeal is made to a selection of texts that seem to illustrate as much. Some key texts include:


Genesis 2-3, since humans require fruit from the tree of life in order to 'live forever', which humans are subsequently forbidden from eating,
1 Timothy 6.16, which says God 'alone has immortality', taken to mean God alone is inherently immortal, while humans are dependent in some way,
Romans 2.7, First Corinthians 15.53-54, Second Timothy 1.10, etc., are understood to indicate that 'immortality' and 'eternal life' are gifts/rewards solely for those who are saved, necessitating a mortality and a temporal life as being the penalty for those who are not saved.




What Is Death?

Because conditionalism and annihilationism are teachings involving 'death', the discussion of course requires defining what 'death' consists of. It is frequently said by advocates of the traditional view ('eternal conscious torment') that 'death' should be defined as 'separation'. Bodily death is separation of the spirit/soul from the body, and spiritual death is separation of the spirit/soul from God. In both cases, the spirit/soul is believed to continue existing consciously in its state of death.

Conditionalists generally criticize this as a redefinition of the word. 'Death', they say, has always been defined as a cessation of life, consisting of a lack of all consciousness, comparable to a dreamless sleep. To support this definition, conditionalists again point to a selection of biblical texts:


Psalm 6.5: For in death there is no remembrance of you; in sheol who will give you praise?
Psalm 115.17: The dead do not praise YHWH, nor do any who go down into silence.
Psalm 146.4: When his breath departs, he returns to the earth; on that very day his thoughts perish.
Ecclesiastes 9.5: For the living know that they will die, but the dead know nothing, and they have no more reward, for the memory of them is forgotten.


This is in addition to the numerous texts throughout the bible that compare death to 'sleep', no matter who it is being described (e.g. 1 Kings 2.10; First Kings 22.40; John 11.11-14; First Thessalonians 4.13f).

With this altogether, conditionalists believe the bible broadly teaches that humans are not innately immortal, in any way, and that their natural end is to die and cease to exist.



Annihilationism

Where conditionalism defines human immortality as conditional upon a right relationship with God, annihilationism is defined as a direct punishment of death from God. Qualitatively, there is no distinction between 'death' and 'annihilation'; the latter word is used solely to clarify just what it is that 'death' consists of.

Again, on a broader level, annihilationists believe the bible teaches that humans who are ultimately unrepentant will suffer death / cessation of existence. Poetic idioms in the Psalms are said to accurately describe a lack of existence, prophetic metaphors are said to capture the essence of a lack of existence, and the 'plain meaning' of basic words are said to describe a lack of existence directly.

The final fate of the unsaved is:


To vanish like smoke (Psalm 37.20)
Like the snail that melts into slime, like the stillborn child that never sees the sun (Psalm 58.8)
Like smoke that is driven away, like wax melts before a fire (Psalm 68.2)
Like a dream when one awakes (Psalm 73.20)
Destroyed, wiped out all remembrance of them (Isaiah 26.14)
Stubble in a burning oven; leaving them neither root nor branch; ashes under the soles of the righteous' feet (Malachi 4.1-3)
Slaying of body and soul (Matthew 10.28)
Eternal punishment (Matthew 25.46)
Death (Romans 6.23)
Eternal destruction (2 Thessalonians 1.9)
Like Sodom and Gomorrah: turned to ashes, and condemned to extinction (2 Peter 2.6)
The second death (Revelation 2.11ff)




Resources

It is widely agreed, even by critics, that the best, complete treatment in favor of conditionalism and annihilationism is The Fire That Consumes, written by Edward Fudge. While it is difficult to speak on behalf of a broad spectrum of belief, Fudge's book is yet considered the masterwork defense of these two beliefs.