In the Forgotten Realms, what happens to a human PC when they die?

Question

In the Forgotten Realms, what happens to a human PC when they die?

Do they have a soul? Where does it go?

Do the resurrected remember their after-death experience?

Answer 1

They rest on the Fugue Plane waiting to be claimed or to strike a deal with a devil to leave it

The Sword Coast Adventurer's Guide provides this information on the afterlife (p. 20):

Most humans believe the souls of the recently deceased are spirited away to the Fugue Plane, where they wander the great City of Judgment, often unaware they are dead. The servants of the gods come to collect such souls and, if they are worthy, they are taken to their awaited afterlife in the deity’s domain. Occasionally, the faithful are sent back to be reborn into the world to finish work that was left undone.

Souls that are unclaimed by the servants of the gods are judged by Kelemvor, who decides the fate of each one. Some are charged with serving as guides for other lost souls, while others are transformed into squirming larvae and cast into the dust. The truly false and faithless are mortared into the Wall of the Faithless, the great barrier that bounds the City of the Dead, where their souls slowly dissolve and begin to become part of the stuff of the Wall itself.

In addition, its section on Asmodeus tells us that the Lord of the Nine Hells can also offer reprieve for those left there (p. 25):

In the beliefs of the people of the North — which coincide with many tales told by dwarves, elves, and others — Asmodeus is Lord of the Ninth, the leader of all devils of the Nine Hells. [...] It’s said that when a soul waits on the Fugue Plane for a deity to take it to its appropriate afterlife, devils approach the soul and offer it a chance at power and immortal pleasures. All a soul needs to do is take one step out of the dust and the milling crowd and put a foot on the first rung of the infernal ladder that represents the hierarchy of the Nine Hells.

[...]

To those not so dedicated, priests of Asmodeus offer the prospect of a reprieve in the afterlife. All souls wait on the Fugue Plane for a deity’s pleasure, which determines where a soul will spend the rest of eternity. Those who lived their lives most in keeping with a deity’s outlook are taken first. Others, who have transgressed in the eyes of their favored god or have not followed any particular ethos, might wait centuries before Kelemvor judges where they go. People who fear such a fate can pray to Asmodeus, his priests say, and in return a devil will grant a waiting soul some comfort.

Answer 2

Their soul gets an express ticket to the Fugue Plane to start with. There they are judged as either a false, a faithless or a faithful. The false are those who betrayed their god. The faithless are those who followed no god. The faithful are those who stayed loyal to their god and are supposed to join them in the afterlife (whether that happens or not depends on what point in history it is, prior to a certain point many of the gods really didn't care much about the souls of their worshipers).