Yes. The frightened condition and the damage caused by the spell are separate.
I read the (amended) spell description as follows:
You tap into the nightmares of a creature you can see within range and create an illusory manifestation of its deepest fears, visible only to that creature.
- The target must make a Wisdom saving throw. On a failed save, the target becomes frightened for the duration.
- At the end of each of the target’s turns before the spell ends, the target must succeed on a Wisdom saving throw
or take 4d10 psychic damage. On a successful save, the spell ends.
Crucially, successful save in the last sentence refers only to the Wisdom saving throw mentioned in the previous sentence; it's describing the corresponding outcome for success to the clause describing failure in the preceding sentence.
Thus, nothing in the spell description makes the psychic damage or the spell duration conditional upon the target being susceptible to the frightened condition. Likewise, if the target was immune to psychic damage but not to the frightened condition, they would still be frightened for the spell's duration.
Since the ongoing psychic damage is independent of the frightened condition, that part of the spell continues to apply if the target is frightened and later becomes immune to being frightened, or has that condition removed.
Contrast this with the description of Fear: (PHB, p. 239)
Each creature in a 30-foot cone must succeed on a Wisdom saving throw or ... become frightened for the duration.
While frightened by this spell, a creature must take the Dash action and move away from you by the safest available route on each of its turns, unless there is nowhere to move. If the creature ends its turn in a location where it doesn’t have line of sight to you, the creature can make a Wisdom saving throw. On a successful save, the spell ends for that creature.
Fear is similar to Phantasmal Killer in that it initially imposes a condition on the target(s), and then has additional effects on each of the targets' turns. But, unlike Phantasmal Killer, Fear makes it clear that the ongoing effects are conditional on the initial condition by using the conditional phrase "While frightened by this spell ... ". There is no such condition applied to the ongoing damage effect of Phantasmal Killer, so it doesn't seem reasonable to read the spell as having the first saving throw mitigate all of the further described effects.
Thus, the target of Phantasmal Killer makes a save when the spell is cast, and is frightened if it fails, and they make a save on each of their turns while the spell is in effect. As soon as they succeed on any of the second kind of saves, the spell ends (and it will also end if the caster loses concentration, or one minute passes).