If the souls of the faithless go to the Wall, the souls of the false get punished or put to work, and the souls of the faithful go to their god's realm (and true atheists go to Nessus for Asmodeus to feed on) then what happens to the souls of people who worshiped Kelemvor?
This is older lore, but you can see from the wiki entry the basic idea.
The petitioners of the City were the servants of Kelemvor and Jergal. The False and the Faithless also inhabited the realm. The servants of Kelemvor and Jergal enacted punishments on the False and acted as a city militia to protect against tanar'ri raids and, on rare occasions, attack the Abyss to punish demons for attacking the city.
For 5e, see a modern interpretation on p20 of the sword coast adventurer's guide.
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.
As the DMG p23 notes
When a creature dies, its soul departs its body, leaves the Material Plane, travels through the Astral Plane, and goes to abide on the plane where the creature's deity resides. If the creature didn't worship a deity, its soul departs to the plane corresponding to its alignment.
So, you go to the fugue plane and probably the city of judgement, to serve Kelemvor however he wishes. This may involve helping judge the dead, or as older lore suggests acting as a city militia.