Abraham and Lazarus were separated from the rich man suffering for his sins. The latter had received his reward in the material things he had sought, craved, and acquired during his mortal lifetime. The gulf Abraham mentions that prevents the wicked from escaping death in the Lake of Fire—and that also keeps the righteous from being burned—is immortality. Those who are immortal will never die because they are composed of spirit like God (Revelation 20:6). Only the saved possess immortality as the gift of God (Romans 2:7).
Conversely, human beings who have not been resurrected or changed to spirit are still physical and subject to corruption and death. They can be consumed by fire because they are composed of flesh and blood. The wicked will reap anguish and wrath, the fiery indignation that will devour the adversary (Hebrews 10:27). For such people, there will be a time of anguish before they die when the fire consumes their bodies. The parable ends with Abraham's words ringing in the rich man's mind and flames of judgment engulfing his body.