What is the biblical basis for the eternal existence of the lost in hell? I've heard particular denominations claim that although hell is definitely a real place and hell itself will burn forever, people who are condemned to hell don't actually exist forever in hell but instead are eventually destroyed, wiped out of existence. They claim that the soul being naturally immortal is a pagan idea, and that only those given eternal life will live forever.

As far as I know most mainline denominations affirm that if you are not saved then you will indeed exist forever in Hell.

My question is, what is the biblical basis to support the doctrine that people who are condemned to hell will actually continue to exist in hell for eternity (as opposed to being annihilated)? Alternatively, what is the biblical support for the soul itself being naturally immortal?
There are indeed many verses that speak of Hell as being eternal although there aren't so many that make the clear that the punishment is also eternal. Here are a couple:

Matthew 25:46 (NIV)
“Then they will go away to eternal punishment, but the righteous to eternal life.”
2 Thessalonians 1:8-9 (NIV)
8 He will punish those who do not know God and do not obey the gospel of our Lord Jesus. 9 They will be punished with everlasting destruction and shut out from the presence of the Lord and from the glory of his might

(Emphasis mine.)
A look at Matthew 25:46 on Blue Letter Bible shows that the same word is used for both usages of &quot;eternal&quot; (and for &quot;everlasting&quot;).  This word is also used in all the instances of &quot;eternal life&quot;, of which there are about 40. Hence, one can safely infer that the two verses I referenced above make it clear that Hell is not a temporary suffering, but that anyone who is sent to Hell will burn forever.