TL;DR
Faithless and False only existed in the forgotten-realms between 1996 and 2021.
Of Deities and Demigods
Back in 1980, TSR published the supplement "Deities and Demigods". It offered a toybox of divine beings, and titled many of them gods, demigods, and heroes. Looking from a religious eye, some of the mythos presented therein don't even have gods in the common meaning of the word. For example, the Arthurian Heroes are offered as a pantheon of divine beings, but none is classed as a god - and nobody would worship them but as a hero. On the other hand, the Cthulhu Mythos is offered as beings, and technically many of them are not even gods but Great Old Ones and Great Ones, and if you follow that pantheon by the letter of the mythos, most of those classed as "gods" don't provide power to their followers.
Of note is that the Arthurian mythos does not show up at all in the Deity list at the end of the book, where gods are given spheres of control, but they are the only stand out in that regard. Also note, that this was way before we properly got the Forgotten Realms. The book does not mention Faithless or Fallen - or afterlives as places to travel to in the first place.
Of Faith and Avatars
In 1996, "Faith and Avatars" was published, the deity sourcebook for the Forgotten Realms. It did bring us Faithless - and the reason they did exist in its intro. Read for yourself how the common folk of the Forgotten Realms gain "their" patron god:
Most folk have a handful of powers that they regularly venerate, only appeasing an unpleasant
power when they are entering or engaged in a situation where that deity holds sway. Most people in the
Realms also eventually settle on a sort of patron deity who they are most comfortable venerating and
who they hold in the greatest reverence. A person’s patron deity is the power that eventually escorts
that person’s spirit from the Fugue Plain, the place where spirits go right after people die, to its afterlife
as a petitioner in the Outer Planes in the realm (or at least the plane) of its patron deity. (Those who firmly deny any faith or have only given lip service most of their lives and never truly believed are known as the Faithless after death. They are formed into a living wall around the City of Strife—Kelemvor, the new lord of the dead, may soon rename it—in the realm of the dead in Oinos in the
Gray Waste and left there until they dissolve. The unearthly greenish mold
that holds the wall together eventually destroys them. The False, those who
intentionally betrayed a faith they believed in and to which they made a
personal commitment, are relegated to eternal punishment in the City of
Strife after their case is ruled upon by Kelemvor in the Crystal Spire
(Kelemvor’s abode in the City of Strife).
p.2-3
So just delivering lip service without actually believing makes people just as Faithless as not having found a patron god yet. Yes, the gods are real, but which of the many gods do you want to worship? Many will follow the local customs, but if there's no fire in your worship and you just follow the motions to silence the community, then you are Faithless. If you found your personal pantheon, but not chose your patron among them, you are still Faithless. Most people will choose their god young, at least for their childhood, as they are brought up in their religious group, but they might go through a phase where they figure out who they really believe in, and to some degree, they might be among the Faithless in that time. And then there are those that deny gods exist, or are worshipable.
The False are those who had found their patron and then betrayed the tenets of the god, be it under pressure or from weakness.
Of Ed Greenwood Presents Elminster's Forgotten Realms
2012 brought us a re-presentation of the campaign setting, and it goes into a tiny bit more detail on when the gods of Faerûn considered you Faithless of False - it's a short list:
The average Faerûnian lives long enough to
worship (or serve through one’s actions) one deity
above all others—though in many cases, which
deity a given person has served most might not be
clear to a dying mortal or anyone else. If a mortal dies before finishing a mission or a task for a
particular deity and it’s a matter he felt strongly
about in life, he could be sent back by that deity,
reborn as another mortal, to try to complete
that task. Otherwise, he ends up in the afterlife
serving the deity most appropriate to his moral
and ethical outlook. Only those who repudiate
the gods (or who as a result of their actions are
renounced by their gods), despoil altars and frustrate the clerical aims of any deity, or never pray
or engage in any form of deliberate worship will
qualify as either Faithless or False.
p.133
The list is short: Disavow your god and you are Faithless until you find a new one to accept you. Be renounced by your god, and you are False. Despoil any altar or frustrate the clerical aims of any deity makes you False. And don't worship at all, and you are Faithless.
Why do they exist? Well, in a strange fashion, it's a tragic end for both heroes and villains: if you destroy the altars and temples of the enemies of your god, you might become branded as False from the opposing side, at least as Elminster puts it. And he ought to know, as he's Mystra's Chosen, and the sage of the Sword Coast.
Outliers from conjecture
Based on non-praying
Non-Outsider Creatures with souls who don't worship (like if they don't have a culture with deities) should be among the Faithless, but that might be a tiny group.
based on ownership of the soul
If you sell your soul to the Abyss or Nine Hells, you also might be considered False for having forsworn the gods, but your soul also wouldn't go to the Fugue Plane in the first place but right to your master to be turned into an inhabitant of their plane and one of their servants.
Of Sword Coast Adventure Guide
The False and Faithless as punished souls still existed in the early 5th edition, but there was not much of a definition for them but their punishment:
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.
p.20
2021 SCAC Errata: False and Faithless Abolished.
In 2021, the Sword Coast Adventure Guide got Errata. These Errata erase the one mention of the False and Faithless, meaning that since that document, there is no more punishment for being a nonbeliever anymore.
The Afterlife (p. 20). In the second paragraph, the
sentence beginning “The truly false and faithless ...” has
been deleted.