Oh yes, the spiritually dead spirit can be tormented for eternity in hell.
Just as the spiritually born again eternally alive spirit has a nature of always loving GOD, the spiritually dead spirit has a nature of always hating GOD.
Let me show you that concept in something that Jesus taught:
Matthew 8:11-12 (ESV)
11 I tell you, many will come from east and west and recline at table with Abraham, Isaac, and Jacob in the kingdom of heaven,
12 while the sons of the kingdom will be thrown into the outer darkness. In that place there will be weeping and gnashing of teeth.”
Matthew 22:13-14 (ESV)
13 Then the king said to the attendants, ‘Bind him hand and foot and cast him into the outer darkness. In that place there will be weeping and gnashing of teeth.’
14 For many are called, but few are chosen.”
Matthew 25:30 (ESV)
30 And cast the worthless servant into the outer darkness. In that place there will be weeping and gnashing of teeth.’
Luke 13:27-28 (ESV)
27 But he will say, ‘I tell you, I do not know where you come from. Depart from me, all you workers of evil!’
28 In that place there will be weeping and gnashing of teeth, when you see Abraham and Isaac and Jacob and all the prophets in the kingdom of God but you yourselves cast out.
Matthew 13:49-50 (NKJV)
49 So it will be at the end of the age. The angels will come forth, separate the wicked from among the just,
50 and cast them into the furnace of fire. There will be wailing and gnashing of teeth."
Outer darkness is a term that symbolically represents Hades or Hell; while the term "gnashing of teeth" is symbolism for "BITTERNESS". In particular it is eternal bitterness towards GOD. There will be no repentance, or mourning over one's sinfulness in hell, only BITTERNESS towards GOD. The term "gnashing of teeth" is in the Greek tense that implies an action that is always ongoing.
We know from Scripture the every human being is made up of three parts:
1 Thessalonians 5:23 (HCSB)
23 Now may the God of peace Himself sanctify you completely. And may your spirit, soul, and body be kept sound and blameless for the coming of our Lord Jesus Christ.
Zechariah 12:1 (HCSB)
1 An Oracle The word of the Lord concerning Israel. A declaration of the Lord, who stretched out the heavens, laid the foundation of the earth, and formed the spirit of man within him.
So the spirit of man is formed at natural birth, but it is spiritually dead in our trespasses, because of the inherited sin from Adam. In other words it is not in tune with the will of GOD, nor is it loving GOD. ONLY when the Holy Spirit has brought our spirit to eternal life, can our spirit be in tune with the will of GOD and truly loves GOD. How then is the term spiritually "dead" applicable? In the same way that faith can be dead:
James 2:26 (NKJV)
26 For as the body without the spirit is dead, so faith without works is dead also.
Many confuse soul and spirit as being the same thing, when Scripture clearly points out they are different, and the Word is sharper than a two edge sword able to divide soul and spirit.
My understanding of the three parts of a human being are:
body = the sum total of all the tangible cells.
soul = the sum total of all the intangible parts of the human psyche (thoughts, memories, emotions, feelings, cognative abilities, etc.)
spirit = the part of man continues on leaving the body upon death, going heaven if one is a believer or hades if one is not.
Interestingly, if you look up soul in the Greek you will find that the Greek word for it is psyche.
So how can we prove that it is spirit of the unbeliever that is tormented forever and ever in hades, and later in hell (the Lake of Fire)?
Matthew 10:28 (NKJV)
28 And 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.
The word "destroy" there does mean "to be totally consumed by" as it would be by fire; BUT NOTICE that HE did not say that about the spirit.
Revelation 20:10 (NKJV)
10 The devil, who deceived them, was cast into the lake of fire and brimstone where the beast and the false prophet are.
And they will be tormented day and night forever and ever.
Revelation 20:15 (NKJV)
15 And anyone not found written in the Book of Life was cast into the lake of fire.
Revelation 14:11 (NKJV)
11 And the smoke of their torment ascends forever and ever; and they have no rest day or night, who worship the beast and his image, and whoever receives the mark of his name."
Matthew 25:41 (NASB)
41 "Then He will also say to those on His left, 'Depart from Me, accursed ones, into the eternal fire which has been prepared for the devil and his angels;
Matthew 25:46 (NASB)
46 "These will go away into eternal punishment, but the righteous into eternal life."
The way I have always taught it, the difference between believers and unbelievers is summed up this way:
Believers are those who want GOD to be part of their lives and they get their wish for ETERNITY, going to be with GOD, which is in heaven.
Unbelievers are those who do NOT want GOD to be part of their lives and they get their wish for ETERNITY, going to where GOD is not, which is in hell, also called the Lake of Fire.