The immolated creature makes a save as normal and, on a fail, suffers the illumination effect but not the fire damage.
The interesting thing in this situation is that the creature is making a dex save to avoid the immolation. Essentially, they are jumping out of the way of the incoming flames. How dexterous they are has no impact on their immunity to fire.
The rules for making saves (PHB 179) states:
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.
The creature is immune to damage from the fire but consider this: if the spell only made the creature cast illumination, might that, in and of itself, be a "risk of harm" (and therefore warrant a saving throw despite the fact that the target is immune to fire damage)?
It would seem the answer is "yes" because the light cantrip requires an unwilling target to make a dex save. By way of example of how light could be detrimental, consider the Gloomstalker ranger's Umbral Sight ability:
While in darkness, you are invisible to any creature that relies on darkvision to see you in that darkness.
So, I'd argue, the creature still needs to make the save against immolation. But what, then, of the illumination effect? If the creature is immune to fire damage, is it immune to the illumination effect as well?
I can easily imagine a situation where a fireproof object is engulfed in flames (one might say "burning") but, because it is fireproof, it is not actually suffering any harm from the blaze.
In the case of an immolated target, it requires only a small amount of imagination to envisage a creature that is alight but not harmed by immolation's flames in the same way. (Think Khaleesi emerging from the burning Dothraki hut in Game of Thrones).
I think the wording of this spell is a little unfortunate because one might read it as the target is doing the illuminating. It's more true to reality (and arguably intended) that the phrase "the burning target" considers the target and the flames around it as a singular combination (that is, the flames and the creature they are engulfing).
So if the target is unable to avoid immolation's flames, it still does continue to shed light as a result of the harmless flames that engulf it even though it is immune to the damage those flames would normally cause.