Phantasmal Killer has been errata'd
http://media.wizards.com/2015/downloads/dnd/PH_Errata_1.1.pdf
Phantasmal Killer (p. 265). The frightened target makes a save at the end of its turns, not the start
So the complete exact text is:
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 start 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.
IMHO DM Ruling is required here.
That successful save refers to any save for that spell or just one of the "avoid the damage" saves?
Since the errata says "the frightened target makes a save" I'd also rule that if the target has the frightened condition removed the entire spell ends.
Please note that this strict ruling makes this 4th level spell sub-part.
Feel free to give Phantasmal Killer some DM love :D
Edit:
After more general considerations and a lenghty discussion with friends I feel the need add this:
Lets consider Vicious Mockery
You unleash a string of insults laced with subtle enchantments at a creature you can see within range. If the target can hear you (though it need not understand you), it must succeed on a Wisdom saving throw or take 1d4 psychic damage and have disadvantage on the next attack roll it makes before the end of its next turn.
If cast on a creature immune to psychic damage should I roll the save?
Yes, because a failed saving throw determines two outcomes, psychic damage and disadvantage on the next attack roll. If the target fails the save it would take no damage but still suffer the disadvantage.
So, back to Phantasmal Killer:
"On a successful save, the spell ends." seems pretty clear to me.
If the save agains fear succeeds the spell ends, if it fails the spell goes on even if the target is immune to the frightened condition. This is brutally RAW.
I very much doubt this is RAI, especially considereing the wording in the errata. In any case, forgoing the first save its a bit too much.
Best Answer
The lore given in the Monster Manual provides an answer
The stat block of any monster is not given in isolation, the Source Book (in this case the Monster Manual) provides lore of each monster presented, and there is no fluff text in 5e (it's rules text all the way down).
This lore segment makes it clear that the paralysis caused by the Mummy's dreadful glare ability in the stat block is caused by the targeted creature being so frightened they are paralysed.
Since the Barbarian cannot be frightened, the Barbarian cannot be frightened into a state of paralysis. Thus the Barbarian is entirely unaffected by the Mummy's Dreadful Glare ability, while the Barbarian is under the effect of their Mindless Rage ability.