There are several mentions of souls in the rules.
What we learn:
- the soul can be separated from the body via Astral Projection (Astral Projection)
- that the soul can leave the body without killing the body (Magic Jar)
- that after death it continues to exist (Raise Dead, Resurrection, True Resurrection, Reincarnation)
- that it is not always free to return once dead (ibid.)
- that it can refuse to return (ibid.)
- that it can perceive who is attempting to resurrect it (ibid.)
- that the soul is separate from the "animating spirit" (Speak With Dead)
- that memories (including languages) are available to the animating spirit in the body separately from the soul (ibid.)
- that memories in a possessed body are not available to the possessing soul (Magic Jar)
- Some Monsters are made from Mortal Souls (MM: Demon, Devil, Genie)
- Vampires' souls are still bound to the body.
Things that being a soulless body apparently prohibits (based upon how Speak With Dead works):
- Learning anything
- Cannot comprehend new information
However, we know from Wraiths and Speak With Dead that Animating Spirits can continue an existence.
And it's pretty clear that ....
- being resurrected, raised, cloned: (which explicitly reunites soul to body in order to function)
- being subjected to Magic Jar as the primary target.
So...
Implications
If your character truly possesses no soul, then it has something functionally similar; a more powerful animating spirit.
It's pretty clear that it cannot use any of Raise Dead, Resurrection, True Resurrection, Reincarnation, Magic Jar, Clone.
It is implied but not clearly stated that Magic Jar would pretty much automatically take over the soulless body.
It's less clear about Astral Projection, but I'd not allow it, either.
It could be argued that the character would be unable to learn, too... at which point they cease being a good potential PC. Free-willed undead are often soulless.
The character likely can be turned as if undead, and may in fact be undead; if it is undead, perhaps it is even unaware it is undead. If not undead, then it's some form of unsouled life.
Supporting Quotes
from the Player's Basic Rules, v. 0.2
Damage types...
Necrotic. Necrotic damage, dealt by certain undead and some spells, withers matter and even the soul.
Astral Projection:
Your astral body resembles your mortal form in
almost every way, replicating your game statistics and
possessions. The principal difference is the addition of
a silvery cord that extends from between your shoulder
blades and trails behind you, fading to invisibility after
1 foot. This cord is your tether to your material body. As
long as the tether remains intact, you can find your way
home. If the cord is cut—something that can happen
only when an effect specifically states that it does—your
soul and body are separated, killing you instantly.
Raise Dead:
You return a dead creature you touch to life, provided
that it has been dead no longer than 10 days. If the
creature’s soul is both willing and at liberty to rejoin the
body, the creature returns to life with 1 hit point.
Resurrection:
You touch a dead creature that has been dead for no
more than a century, that didn’t die of old age, and that
isn’t undead. If its soul is free and willing, the target
returns to life with all its hit points.
Speak With Dead:
Until the spell ends, you can ask the corpse up to five questions. The corpse knows only what it knew in life, including the languages it knew. Answers are usually brief, cryptic, or repetitive, and the corpse is under no compulsion to offer a truthful answer if you are hostile to it or it recognizes you as an enemy. This spell doesn’t return the creature’s soul to its body, only its animating spirit. Thus, the corpse can’t learn new information, doesn’t comprehend anything that has happened since it died, and can’t speculate about future events.
True Resurrection:
You touch a creature that has been dead for no longer than 200 years and that died for any reason except old age. If the creature’s soul is free and willing, the creature is restored to life with all its hit points.
Additional Supporting Quotes from the PHB
Clone:
This spell grows an inert duplicate of a living creature
as a safeguard against death. […] It remains
inert and endures indefinitely, as long as its vessel
remains undisturbed.
At any time after the clone matures, if the original
creature dies, its soul transfers to the clone, provided
that the soul is free and willing to return. The clone is
physically identical to the original and has the same
personality, memories, and abilities, […]. The original creature's physical
remains, if they still exist, become inert and can't thereafter be restored to life, since the creature's soul
is elsewhere.
Magic Jar:
Your body falls into a catatonic state as your soul leaves it and enters the container you used for the spell's material component. […] The only action you can take is to project your soul up to 100 feet out of the container, either returning to your living body (and ending the spell) or attempting to possess a humanoids body.
[…] [possession] The target must make a Charisma saving
throw. On a failure, your soul moves into the target's
body, and the target's soul becomes trapped in the
container.
Reincarnate:
You touch a dead humanoid or a piece of a dead humanoid. Provided that the creature has been dead no longer than 10 days, the spell forms a new adult body for it and then calls the soul to enter that body. If the target's soul isn't free or willing to do so, the spell fails.
Transitive Planes:
The Astral Plane is the realm of thought and dream, where visitors travel as disembodied souls to reach the planes of the divine and demonic.
from the Monster Manual
Demon:
Other demons (such as manes) are created from mortal souls shunned or cursed by the gods, or which are otherwise trapped in the Abyss.
and
As long as the demon remains in possession of its host, the soul of that host is in danger of being dragged to the Abyss with the demon if it is exorcised from the flesh, or if the host dies.
Devil:
Lemures. The lowest form of devil, lemures are the twisted and tormented souls of evil and corrupted mortals.
Genie:
A genie is born when the soul of a sentient living creature melds with the primordial matter of an elemental plane. Only under rare circumstances does such an elemental-infused soul coalesce into a manifest form and create a genie.
A genie usually retains no connection to the soul
that gave it form. That life force is a building block that
determines the genie's form and apparent gender, as
well as one or two key personality traits.
Wraith:
When a mortal humanoid lives a debased life or enters into a fiendish pact, it consigns its soul to eternal damnation in the Lower Planes. However, sometimes the soul becomes so suffused with negative energy that it collapses in on itself and ceases to exist the instant before it can shuffle off to some horrible afterlife. When this occurs, the spirit becomes a soulless wraith-a malevolent void trapped on the plane where it died. Almost nothing of the wraith's former existence is preserved; in this new form, it exists only to annihilate other life.
Other Dribs and Drabs
Vampire:
Afflicted characters have the following options:
[...]
Vampirism. These characters can be cured either by a wish spell cast by a fellow PC (wish is not available as a spellcasting service), or they may be slain and returned to life via raise dead–at the normal cost of 1250gp.
(Adventurer's League FAQ update 5-19-2015)
