As mentioned in a previous article, the Epic of Gilgamesh depicts Gilgamesh acquiring a plant granting immortal youth, which is lost due to a snake. Genesis 3 depicts Adam and Eve in the presence of a tree granting immortality, which is lost due to a snake. The simple explanation for the overlap in imagery is either that both stories are drawing from the same well of symbolism, or perhaps that the author of Genesis 2-3 consciously borrowed the imagery from Gilgamesh and remixed them to suit his own purposes.
Western Christianity has traditionally seen in Genesis 3 a 'fall' of humanity. In disobeying God's command, Adam's very being became corrupted, and Eve with him. Their 'fall' removed them from God's grace, and they proceeded to pass their new 'sin nature' onto their descendants, namely, all of humanity. Because every person is descended from Adam and Eve, every person is conceived in a 'fallen' state outside of God's grace. Hence, because of Adam's sin, the immortal soul of every individual is condemned to eternal torment in hell by default of existing.
It is generally agreed among critical scholars that this theological view goes back to Augustine, who built it by relying on a Latin mistranslation of Romans 5. In that text, Paul uses Adam as a typological figurehead for sinful humanity, but he does not attribute Adam's sin to the rest of humanity; instead, Paul explicitly holds every individual responsible for their own sins ('just as sin came into the world through one man, and death came through sin, so death spread to all because all have sinned').
Genesis 2-3 flatly contradicts the notion of an 'immortal soul'. When the first man is created, his body is filled with breath, creating 'a living soul' (nephesh hayyah). This is in contrast to a 'soul' that is 'dead' (cf. Ezekiel 13.19; 18.4,20; James 5.20). The story in question plainly depicts humanity as mortal by nature, since the Tree of Life is present from the moment of the first man's creation, and this tree's very purpose is to provide immortality (3.22), and God bars humanity from eating the tree's fruit (3.23).
Any concept of humans spending their existence in either heaven or hell is absent from this story. The focus is on this life, and the curses pronounced over Adam, Eve, and the snake are very clearly physical, not metaphysical.
The snake (nowhere identified in the text as the satan) is condemned to slither along the ground and eat dust. John Walton, in The Lost World of Adam and Eve, points to Egyptian incantations to provide context here. In ancient Egyptian religion, snakes were obstacles on the road through the afterlife; people were taught incantations to recite when walking this road, so they could command the snakes to stay slithering on the ground, unable to raise their heads and poise to bite the traveler's legs.
Words to be spoken: "This is the fingernail of Atum, which was on the backbone of Nehebu-kau and which brought to an end the strife in Hermopolis. Fall, roll up!"
Words to be spoken: "Back with thee, hidden snake! Hide thyself! Thou shalt not make King Unis see thee. Back with thee, hidden snake! Hide thyself! Thou shalt not come to the place where King Unis is, lest he tell that name of thine against thee: Nemi, the son of Nemit. The servant of the Ennead fell into the Nile. Turn about, turn about! O monster, lie down!"
Pritchard, Ancient Near Eastern Texts Relating to the Old Testament, 326
Adam's creation from the ground is actually the start of a theme that will be touched on again in the 'J source' in Genesis. Adam is made from the ground (2.7), he is made to work the ground (2.15), and when he disobeys God's command he is cursed so that he must struggle in working the ground (3.17-18) until he eventually dies and his body returns to the ground (3.19).
As far as Genesis 2-3 is concerned, the 'fall' of humanity is a cursed relationship with the ground, not a 'sin nature'.
It is sometimes claimed that Genesis 2-3 has more going on in the background than is immediately obvious. The way the story is told — God gives Adam a commandment, Adam breaks the commandment, Adam 'dies' by being exiled from his divinely-appointed land to the east — is said to be a direct parallel to the Babylonian exile. In this reading, Adam's story is Israel's story, from their receiving the commandments of the Torah, to their repeated violation of those commandments, to their exile from their divinely-appointed land to Babylon in the east. The snake is connected to idolatry, Eve is even connected to sexually immoral acts, and Jerusalem is connected with Eden. (See, e.g., George & George, The Mythology of Eden.) There is some merit to this. For example, it explains how Adam died 'in the same day' as when he sinned; to ancient Israel, exile is death (cf. Deuteronomy 28-30), and Adam's exile lost him the immortality of the Tree of Life thus guaranteeing his death. But does this theory work as a whole?
I see little evidence for this. As I have already noted, it is difficult to envisage the serpent as a symbol of the idolatry that led Judah astray before the exile. The serpent is nowhere a symbol of Baal, and serpent worship does not feature in the Old Testament's depiction of the monarchical period apart from the cult of Nehushtan, something already eliminated by Hezekiah over 100 years before the exile (2 Kgs 18.4). Furthermore, the serpent in Genesis 3 is not actually worshipped. Again, the crime of the first humans was in seeking the knowledge of good and evil by eating the forbidden fruit, and it is difficult to relate that to Judah's situation before the exile. Nowhere does the Old Testament represent the fall of Judah as being due to seeking after forbidden wisdom. We have, moreover, already seen that the case for the identification of Eden with Jerusalem in Gen. 2.10-14 is extremely weak, and the theme of expulsion is characteristic of other J stores in Genesis 1-11 which clearly have nothing to do with the Babylonian exile (cf. Gen. 4.13-16; 11.7-9). Moreover, the expulsion from Eden sounds final and irreversible, which is in contrast to what the Old Testament elsewhere says about the Babylonian exile. It seems better, therefore, to regard the story as relating simply to the first humans and the human situation generally, as its surface meaning suggests, rather than to Judah and the exile.
Day, From Creation to Babel, 46-47
The story of Adam and Eve being tempted by a snake, which results in their loss of a tree imparting immortality, is a remix of common images in Mesopotamian myths. The snake is not the devil; it is, as the narration states, only an animal of the field. The curses on the snake and Eve are etiologies, showing why humans hate snakes, why women suffer so much pain giving birth. Despite one popular interpretation, the story is not an allegory for the Babylonian exile. Likewise, neither does it present humanity is inherently immortal in any sense. The 'fall' of humanity was not the manifestation of a 'sin nature', but of Adam's broken relationship with the earth from which he was made.