First we need to understand mechanically how immunity to a condition actually functions when a saving throw is involved. This can be observed by the following situation:
A Wizard casts Sunburst on an Ochre Jelly;
On a failed save, a creature takes 12d6 radiant damage and is blinded for 1 minute.
An Ochre Jelly is immune to the Blinded condition, but a saving throw is still called for.
Undead and oozes have disadvantage on this saving throw.
The Ochre Jelly is even making the saving throw with disadvantage, but is still unable to be affected by the blinding effect of the spell due to its immunity, which indicates that a condition imposed by a spell as the result of a saving throw has no effect.
If a Paladin casts Thunderous Smite (PHB, p. 282) on the same Ochre Jelly;
Additionally, if the target is a creature, it must succeed on a Strength saving throw or be pushed 10 feet away from you and knocked prone.
This is another example of the one part of the effect of the spell simply being ignored while a saving throw is required to be rolled because the Ochre Jelly is immune to the condition that the spell inflicts.
This is all to say that immunity to a condition would just mean that if this condition were to affect it, it would do nothing instead.
The Fear spell reads as such:
Each creature in a 30-foot cone must succeed on a Wisdom saving throw or drop whatever it is holding and become frightened for the duration.
The target still makes a saving throw as called for, but regardless of whether it fails or passes, it cannot be frightened. Any changes made by a portent or Cutting Words would be wasted if the characters were unaware of the immunity. However, RAW, this does also mean that a creature with immunity to the fear condition would still drop whatever it is holding as that is specifically not a part of the frightened condition, but an effect of the spell itself.
Saving Throws in the PHB pg. 179 reads as such:
A saving throw—also called a save—represents an
attempt to resist a spell, a trap, a poison, a disease, or
a similar threat. You don’t normally decide to make a
saving throw; you are forced to make one because your
character or monster is at risk of harm.
And further in the PHB on pg. 205
Many spells specify that a target can make a saving
throw to avoid some or all of a spell’s effects. The spell
specifies the ability that the target uses for the save and
what happens on a success or failure.
Other than the slightly loose wording on page 179, nowhere does it indicate that a saving throw is ever optional or is not to be rolled regardless of the outcome of the effect, and a creature is unable to even willingly fail a saving throw as per Jeremy Crawford's response to this question.
Only the possible reading of "can" from page 205 would indicate a possibility of having an option of a saving throw, but this can be negated by the specific wordings of spells, such as the following:
A better example to consider for this question that has no extra clauses in the effect would be Blindness/Deafness
You can blind or deafen a foe. Choose one creature that you can see within range to make a Constitution saving throw. If it fails, the target is either blinded or deafened (your choice) for the duration.
RAW, "to make a Constitution saving throw" indicates that regardless of the Ochre Jelly's immunities, the save is rolled. It does not state "unless the target is immune" or "if the condition imposed by this spell would have no effect on a success". On a fail, the condition simply does not take effect because of the immunities of the target. Additionally, this would also indicate that the spell is continuing for the duration or until a successful save is rolled.
At the end of each of its turns, the target can make a Constitution saving throw. On a success, the spell ends.
Obviously as has been stated before, a GM may decide to rule it however they wish in the interest of time or enjoyment as they are the final word at their particular table, and I am sure that many would agree it would make the most sense to not make needless rolls or have your players waste resources; but as everything is written and strictly speaking a roll would need to be made regardless of immunity to the effect of a spell that calls for a saving throw because it never says otherwise.