If the "torment" (3600 ὀδυνάω odunao) is the classic idea of hell-fire, the Rich Man would ask for a lot more than a little water to cool his tongue. Odunao here refers to mental anguish (Luke 2:48; 16:25; Acts 20:38). Its antithesis is parakaleo 'to be consoled for sorrow' like Lazarus (Lk. 16:25).22 "So it was that the beggar died, and was carried by the angels to Abraham's bosom. The rich man also died and was buried.
23 "And being in torments (931 βάσανος basanos) in Hades, he lifted up his eyes and saw Abraham afar off, and Lazarus in his bosom.
24 "Then he cried and said, `Father Abraham, have mercy on me, and send Lazarus that he may dip the tip of his finger in water and cool my tongue; for I am tormented (3600 ὀδυνάω odunao) in this flame (5395 φλόξ phlox).'
25 "But Abraham said, `Son, remember that in your lifetime you received your good things, and likewise Lazarus evil things; but now he is comforted (3870 παρακαλέω parakaleo) and you are tormented (3600 ὀδυνάω odunao). (Lk. 16:22-25 NKJ)
Christ chose the word "torment" (931 βάσανος basanos) to describe the Torment in Hades:
This "torment" is the kind that brings truth to the surface, resulting in "anguish" and "sorrow" (3600 ὀδυνάω odunao cp. Acts 20:38) for sin. The imagery of purging from sin is common in the Old Testament (Zech. 13:9; Ps. 51:7; 66:10; Isa. 1:25; 4:4; 6:7; 48:10; Ezek. 24:13; Dan. 11:35):Meaning: 1) a touchstone, which is a black siliceous stone used to test the purity of gold or silver by the colour of the streak produced on it by rubbing it with either metal 2) the rack or instrument of torture by which one is forced to divulge the truth.-Strong's Concordance
In Luke 16:24 the "flame" (5395 φλόξ phlox) causes the "agony" or "anguish" (3600 ὀδυνάω odunao). The Rich Man is being purged (Mal. 3:3; Job 23:10; Zech. 13:9 cp. 1 Pet. 1:7) by the "flame of God's inspection": "His eyes like a flame" (Rev. 1:14; 2:18; 19:12). The "flame" is the "visible aspect of a fire that springs upward and is usually orange"(Logos Bible Sense Lexicon). God is revealed in the "flame" (Ex. 3:2; Judges 13:20; Isa. 66:15 LXX; Acts 7:30; 2 Thess. 1:8).2 "But who can endure the day of His coming? And who can stand when He appears? For He is like a refiner's fire And like launderer's soap.
3 He will sit as a refiner and a purifier of silver; He will purify the sons of Levi, And purge them as gold and silver, That they may offer to the LORD An offering in righteousness. (Mal. 3:2-3 NKJ)
The "torments of Hades" are designed to bring the true person to the surface, liberate their free will from all bondage to sin and delusion. After suffering for their sins "judged according to men in the flesh", if they choose to believe the gospel preached to them in Hades they will "live according to God in the Spirit" apart from any torment, eagerly waiting for the Second Coming of Christ when they will rise with the dead in Christ:
Is The Rich Man and Lazarus Story About Hell Prophecy Parable or both? | End Time News
34 All these things Jesus spoke to the multitude in parables; and without a parable He did not speak to them, 35 that it might be fulfilled which was spoken by the prophet, saying: "I will open My mouth in parables; I will utter things kept secret from the foundation of the world." (Matt. 13:34-35
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[Compare the similar wording in 1 Peter 3:18 θανατωθεὶς μὲν σαρκί, ζῳοποιηθεὶς δὲ πνεύματι "put to death in the flesh, but quickened by the Spirit". This implies the "trial/judgment/torment" in hades ends" when they repent and "live according to God in the Spirit" (ζῶσιν δὲ κατὰ θεὸν πνεύματι). Their "spirit" quickened is now abiding surrounded by the Holy Spirit (Compare ἐν ᾧ καὶ "in which also" 1 Pet. 3:19) shielded from any torments.]6 For this reason the gospel was preached also to those who are dead, that they might be judged according to men in the flesh, but live according to God in the spirit. (1 Pet. 4:6 NKJ)
After death is a trial, a judgement, so Christ was offered to bear their sins to make possible they pass the Trial by believing the gospel preached to the dead (1 Pet. 6:4) and then eagerly wait for Christ's return, for salvation.
27 And as it is appointed for men to die once, but after this the judgment,[1]
28 so Christ was offered once to bear the sins of many. To those who eagerly wait for Him He will appear a second time, apart from sin, for salvation. (Heb. 9:27-28 NKJ)