Yes, souls that are not saved are destroyed in the lake of fire. This is called the "second death" (the first death being the death of the body).
"I [also] saw the dead, great and small; they stood before the throne, and books were opened. Then another book was opened, which is [the Book] of Life. And the dead were judged (sentenced) by what they had done [their whole way of feeling and acting, their aims and endeavors] in accordance with what was recorded in the books. And the sea delivered up the dead who were in it, death and Hades (the state of death or disembodied existence) surrendered the dead in them, and all were tried and their cases determined by what they had done [according to their motives, aims, and works]. Then death and Hades (the state of death or disembodied existence) were thrown into the lake of fire. This is the second death, the lake of fire. And if anyone’s [name] was not found recorded in the Book of Life, he was hurled into the lake of fire." --Revelation 20:12-15 (Amplified Bible)
"But as for the cowards and the ignoble and the contemptible and the cravenly lacking in courage and the cowardly submissive, and as for the unbelieving and faithless, and as for the depraved and defiled with abominations, and as for murderers and the lewd and adulterous and the practicers of magic arts and the idolaters (those who give supreme devotion to anyone or anything other than God) and all liars (those who knowingly convey untruth by word or deed)—[all of these shall have] their part in the lake that blazes with fire and brimstone. This is the second death." --Revelation 21:8 (Amplified Bible)
Spirit is immortal; it cannot die. It returns to its source upon physical death--but it is also impersonal. There is no part of "you" that makes you "you" that survives as spirit. Spirit is merely the life force that makes the inanimate matter that composes our bodies alive.
"Then shall the dust [out of which God made man’s body] return to the earth as it was, and the spirit shall return to God Who gave it." --Ecclesiastes 12:7 (Amplified Bible)