The Paladin's Aura of Courage prevents the frightened condition in anyone within 10' of the paladin. No saving throw is needed.
The Pit Fiend's Fear Aura behaves thusly:
Any creature hostile to the pit fiend that starts its turn within 20 feet of the pit fiend must make a DC 21 Wisdom saving throw, unless the pit fiend is incapacitated. On a failed save, the creature is frightened until the start of its next turn. If a creature’s saving throw is successful, the creature is immune to the pit fiend’s Fear Aura for the next 24 hours. (MM, Pit Fiend)
Expected Scenario:
The party encounters the Pit Fiend while they are within 10' of the Paladin. The Pit Fiend tosses a fireball (an at-will spell) at the party. It then closes to within 20' of the party (it may not realize that there's a Paladin in there) with this intention: to break up the attackers by making all or most of them frightened and then proceed beat at least one of them down on the subsequent turn with its four attacks, or, maybe toss another fireball to soften them up more. If they just stand there, frightened. The fear aura fails, of course, and no PCs are frightened.
The party realizes that staying in "fireball formation" will help the Pit Fiend cook them all so they decide to spread out. As each party member leaves the Paladin's aura, do they (if they are, or move, within the 20' radius of the pit fiend's aura) then have to make the saving throw against the pit fiend's fear aura?
The reason I ask: the first time they were subjected to the fear aura they were not affected by it. One can argue that this is equivalent to, or even better than, a successful saving throw. (see the bolded text in the citation above)
But they never rolled the dice.
What I need to rule on as the DM
Do the PCs other than the Paladin need to roll that saving throw against the Pit Fiend's aura as they leave the Paladin's aura, or, have they acquired immunity from the interaction between the aura of the Pit Fiend and the aura of the paladin that failed to frighten them?
While I suspect that they have to roll the save, I may be missing a detail here.
What problem am I trying to solve?
The players I DM for just got to level 10, there is a Devotion Paladin in the party, and I foresee that in the next five or six sessions they may make a decision that triggers, or leads to, a Pit Fiend encounter - or they may not. I want to be ready for that encounter if their choices take them in that direction.