Raise Dead — No. The spell description, at the end of the second paragraph, states:
The spell can't return an undead creature to life. (PHB, p. 270)
Revivify — Probably Not. "[A] creature that has died within the last minute ... returns to life with 1 hit point." (PHB, p. 272) Both Animate Dead and Create Undead have a casting time of 1 minute, so their targets could not become undead before Revivify is ineffective. Similarly, many undead monsters take well over a minute to make (e.g., liches, mummies, vampires). However, there appear to be other ways to create undead in less than a minute, for example, the Finger of Death spell and death tyrant's Negative Energy Cone raise zombies the next turn, a shadow dragon's Shadow Breath creates a shadow the same turn it kills a humanoid, and the wraith has the Create Specter ability. These cases may require DM adjudication. One way could be by Contest (DMG, p. 238; PHB, p. 174) for the soul to take over its undead body due to the opposing magics involved. See generally, Magic Jar spell description (PHB, p. 257). But see, Discussion below.
True Resurrection — Maybe. The spell states, "If the [dead] creature's soul is free and willing, the creature is restored to life ... the spell can even provide a body if the original no longer exists." (PHB, p. 284) This suggests that incorporeal undead are resurrected if their soul is free, as the spell would simply create a new body. For corporeal undead, the question is whether True Resurrection can supplant the undead force with a free soul. (If permitted, this could be done by Contest. See above.)
Ultimately, the DM must decide whether being undead somehow makes a soul not free and thus unable to return to life. See Discussion below.
Discussion
Although not explicit in its description, Revivify probably does not work on an undead creature without a body given that it requires "touch[ing] a creature that has died within the last minute." (PHB, p. 272) According to the Monster Manual, a specter is a "bodiless spirit" (p. 7) and has Incorporeal Movement (p. 279) so True Resurrection may work by creating a new body. However, the specter's description suggests that it is the soul itself—"Some are spawned when ... the touch of a wraith rips the soul from a living body" and "its only end the oblivion that comes with the destruction of its soul" (Ibid.)—implying that the soul is not free to be restored to life.
If a creature is killed by the spell Finger of Death, "it rises at the start of [the caster's] next turn as a zombie." (PHB, p. 241) According to the MM, "Once turned into a zombie, a creature can't be restored to life except by powerful magic, such as a resurrection spell." (MM, p. 315) Reading way too much into this, "resurrection" here is not italicized so it is not the specific Resurrection spell. However, it does say "powerful magic" and a 3rd-level spell like Revivify is not powerful magic, so I don't think it would work; but, True Resurrection is the most powerful resurrection spell and might depending on your DM's interpretation.
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.
Best Answer
A dead lich may be resurrected if its phylactery is also destroyed
When discussing whether true resurrection can raise a lich, there are 3 distinct cases that must be considered.
Undead/Not dead lich - cannot be targeted by true resurrection
True resurrection requires a dead creature to be the target of the touch. A lich is an undead creature, but it is not dead. Jeremy Crawford confirms this:
It is thus not a valid target for true resurrection.
So, using true resurrection will only work on a lich that has been killed. The effectiveness of the spell in this case will be determined by the state of the lich's phylactery.
Dead lich, phylactery intact - cannot be resurrected
If you kill the lich (before destroying the phylactery) then use true resurrectionit likely will not be able to come back to life because:
A soul that is trapped forever in a container is not "free and willing" which means that the soul does not meet this condition of true resurrection:
Since it does not meet that condition, the lich cannot be restored to life by true resurrection while its phylactery remains intact and active.
Dead lich, phylactery destroyed - can be resurrected as non-undead creature
If you kill the lich (after destroying the phylactery) then use true resurrection it will return to life as as the creature it originally was.
Per the description of true resurrection (after errata):
Assuming that the soul is now willing after being freed from the phylactery this means that the (former) lich can now be resurrected to its non-undead form.