Per the Lich's Soul Sacrifice description, both the body and soul of the imprisonment spell's target are trapped inside the phylactery. The description also states that this imprisoned "creature" is destroyed in 24 hours.
the lich uses the [imprisonment] spell to magically trap the target's body and soul inside its phylactery…. A creature imprisoned in the phylactery for 24 hours is consumed and destroyed utterly, whereupon nothing short of divine intervention can restore it to life.
p. 203 Monster Manual 5e (emphasis mine)
If we are to interpret the term "creature" as referring to both the body and soul of the imprisoned entity, then the soul is destroyed utterly.
If the soul no longer exists, does it have any afterlife experience?
Additional thoughts:
Might a person voluntarily offer their soul to a lich to avoid judgment for their False or Faithless soul?
The last bit of the Soul Sacrifice description does state that divine intervention can restore a destroyed creature to life, so it likely would not be outside of the power of Kelemvor to recover a soul attempting to avoid his judgment.
Best Answer
Short answer: Yes. A soul consumed in a phylactery avoids any afterlife, and once destroyed after 24 hours it is gone completely.
With the religious structure in Forgotten Realms, any type of afterlife is depending on the Deity the character worships. To the point that if the character does not worship any kind of Deity in the Forgotten Realms setting, then divine spells to bring the character back from the dead are not effective, and the character's soul is supposed to sit in a kind of limbo for the rest of existence.
Also, the Imprisonment spell traps both body and soul, so it's better to think of the character being transported to another realm. With both body and soul intact and still connected, the character should still have awareness of their surroundings. In this case, their existence is now inside the Lich's phylactery.
There is no source that describes any specifics about what a creature experiences inside a phylactery. Whatever the character perceives in there is up to the creativity of the DM and the player.
However, since the soul has to be free to be delivered to their Deity to experience an afterlife, the character wouldn't experience that. While in the phylactery, the character doesn't experience an afterlife. After that 24 hour period, the soul is destroyed, not transported to their deity to experience an afterlife.