The MM entry for a Lich mentions destroying a phylactery is a difficult endeavor but the only way to prevent a Lich from rising again and again.
Does destroying the phylactery destroy the soul within? If not, is there existing lore explaining what happens to a Lich's soul when its phylactery is destroyed?
Lich phylacteries in general can permanently destroy or consume souls fed to them by the required ritual, but presumably the Lich's soul is not one of them so long as the phylactery is intact.
D&D 5e answers preferred, but mechanics and lore from other related systems will be noted for guidelines if nothing in 5e exists on the topic.
Best Answer
No
The simple answer is that destroying the phylactery does not destroy the soul contained inside it, because no game rule in D&D 5th edition asserts that it would.
The lore in the Monster Manual, p.203 suggests that the phylactery is not necessary to protect the soul from injury, but rather to prevent it from passing to the afterlife:
Destroying the phylactery doesn't kill the lich, but only makes it possible to kill the lich, since with the phylactery intact, a slain lich simply re-constitutes at their phylactery:
What exactly happens to the soul when the phylactery is destroyed is a mystery (see my answer to Can a Lich create a new phylactery?).