The Sorcerer's Empowered Spell metamagic option reads
Empowered Spell
When you roll damage for a spell, you can spend 1 sorcery point to reroll a number of the damage dice up to your Charisma modifier (minimum of one). You must use the new rolls.
While the spell Immolation reads (emphasis added):
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.
If a sorcerer with +5 CHA uses spends 1 sorcery point for Empowered Spell on Immolation to re-roll 3 of the 8d6 and the target fails its save on its next turn can the Sorcerer re-roll an additional 2 dice of the 4d6 damage because they have not yet re-rolled the maximum dice (5) allowed by using Empowered Spell?
This question might include whether Empowered spell should be read as:
"When you roll damage for a spell, you can spend 1 sorcery point to reroll a number of the damage dice [from that damage roll] up to your Charisma modifier"
or
"When you roll damage for a spell, you can spend 1 sorcery point to reroll a number of the damage dice [for that spell] up to your Charisma modifier"
This question is about a single use of Empowered Spell.
Best Answer
Yes, that is supported by a Sage Advice ruling on Scorching Ray
Empowered Spell:
Note that it says "the damage dice" [for the spell]. If it had said "When you roll damage for a spell, you can spend 1 sorcery point to reroll a number of those damage dice", then that would be an unambiguous statement that the only dice that may be rerolled are specifically those which triggered the Empowered Spell Ability. However, since it only says "the damage dice", it leaves open the possibility that for a spell that has multiple sets of damage dice, one set of dice may be the trigger for empowered spell and another set may be the ones actually rerolled, so long as those are also damage dice of the spell.
For a look at how this has been adjudicated with another spell, see this Sage Advice about Scorching Ray, where it clearly says that the Empowered Spell ability may apply to different rays.
Note that these different rays are each different attacks, and that they must therefore be resolved in a temporal order. If the first ray hit and thus did damage, that could trigger the Empowered Spell ability, which could then be used to reroll dice on that ray or a second ray or both (according to Sage Advice), including on a ray which had not yet been determined to hit or miss. This is permissible even though the damage occurs at two different points in time, even though the damage that triggered the Empowered Spell is not necessarily the damage that was rerolled, and even though the damage to be rerolled does not yet definitively exist, because both sets of damage dice are damage from the same spell.
Moving to your question about Immolation, this is still damage from the same spell, and although the damage on the second turn may not happen (due to a successful save), the same can be said of the potential damage from a second Scorching Ray (that may not hit). The only difference between the two examples is that the Scorching Ray damage dice are rolled at different times but all on the same turn, whereas the Immolation dice are ruled over multiple turns.
The one thing the Sage Advice does not tell us is whether one may wait to see the results of a roll before deciding which other dice to reroll. For example, if I have a Cha +3, three dice to reroll, and some of the rerolls are being used on dice that did not trigger the Empowered Spell, may I decide which ray(s) to spend my dice on after I see whether they hit, or after I see whether what their damage is? Or, do I have to assign all the rerolls all at once, at the moment the metamagic ability is used? See this question.