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I am already aware of the question "Does a creature that is immune to all outcomes (conditions, damage, etc.) of an effect still make a saving throw?" and also "Is a creature immune to a spell's damage type immune to the spell's other effects?" but I'm unsure how to tell if effects of a spell are "linked" or not. For example there have been the questions "How does a ghost's Horrifying Visage aging interact with immunity to the frightened condition?" and "Does immunity to fear prevent a mummy's Dreadful Glare from paralyzing a character?" where it is contentious whether or not the effects are linked and even whether or not the target makes a saving throw at all.

And thus I am wondering about the immolation spell which states:

[...] The target must make a Dexterity saving throw. It takes 8d6 fire damage on a failed save, or half as much damage on a successful one. On a failed save, the target also burns for the spell’s duration. The burning target sheds bright light in a 30-foot radius and dim light for an additional 30 feet [...]

Assuming the target is immune to fire damage, what happens when they are targeted by immolation? Do they shed light if the fail the saving throw? Do they make a saving throw at all?

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.

Answer changed to reflect further understanding.

The target is uninjured, but surrounded by magical fire that produces light, if they fail the saving throw.

Flames wreathe one creature you can see within range. The target must make a Dexterity saving throw. It takes 8d6 fire damage on a failed save, or half as much damage on a successful one. On a failed save, the target also burns for the spell's duration. The burning target sheds bright light in a 30-foot radius and dim light for an additional 30 feet. At the end of each of its turns, the target repeats the saving throw. It takes 4d6 fire damage on a failed save, and the spell ends on a successful one. These magical flames can't be extinguished by nonmagical means.

In the example situation, because the target is immune to fire damage, it cannot be harmed.

However, the spell forms magical fire around the target, which stays if they fail the saving throw. It is likely this magical fire is the source of the light the spell mentions.

Which means as long as the target fails the saving throws, they will be surrounded by a light source.