Fueled Spellfire with Divine Smite and Critical Hit.

SPELLFIRE ADEPT:

Fueled Spellfire. Once per turn, when a spell you cast deals Radiant damage, you can expend up to two Hit Point Dice, roll them, and add the total rolled to one damage roll of the spell.

I think that wording means those are not really intended to count as damage dice and therefore shouldn't be doubled on a critical hit.

That said, since these are Hit Point Dice we're talking about, if someone wanted to burn twice as many of them on a critical I'd probably let them.

Based on Divine Smite adding damage to the attack and Fueled Spellfire adding the total of its effect to the damage roll, I'd have to agree that the dice of Fueled Spellfire aren't damage dice for the attack, but rather function as a damage bonus (the same way a stat modifier or magic enhancement would).

I suspect you are right.

But then, when you cast Divine Smite, don't you also roll extra 2d8 radiant and add the total to one damage roll of your weapon?

It is a weird wording, but I hoped that the "add the total rolled to one damage roll of the spell" was meant just for avoiding the extra damage to be added to spells that do multiple damage rolls per round, like Divine Favor. 
I wish they had been a bit clearer TBH. 

Divine Smite:The target takes an extra 2d8 Radiant damage from the attack.

Fueled Spellfire:You can expend up to two Hit Point Dice, roll them, and add the total rolled to one damage roll of the spell.

The difference is subtle, but one adds dice to the attack, one adds the total of the roll to the damage. This distinction also shows in similar effects, like Sneak Attack or Hunter's Mark:

Sneak Attack:Once per turn, you can deal an extra 1d6 damage to one creature you hit with an attack roll.

Hunter's Mark: You deal an extra 1d6 Force damage to the target whenever you hit it with an attack roll.

Both of these effects add extra dice, where Fueled Spellfire calculates the total before adding to the damage roll. That's how I'd read it anyway, it's similar enough that I could see it going the other way (and I don't think the extra damage is going to be a huge deal, to be honest, it's only up to 2d12 at most if you're multiclassed in Barbarian and choosing to cast a spell). Plus it consumes a resource.

Not sure if it helps, but I had a doubt related to Fire's Burn, Frost's Chill and Critical Hits, in case some similarity can be found.

Also, from the Dev:

@quadhund If I make 1 attack with a longsword as a part of casting Green Flame Blade, while having the Hex spell affecting the target, and then I use Divine Smite once the attack hits, how many damage rolls are there?
@JeremyECrawford When something in the game (Sneak Attack, Divine Smite, hex, etc.) causes your attack/spell/etc. to deal extra dice of damage, those dice are added to the damage the effect is already dealing, if any. It's one big damage roll, extra damage included. 
@DaveWil33 Would only be a second damage roll if there was something locked behind a save like poison damage, yes?
@JeremyECrawford If an attack has a damage roll but also a second damage roll (not extra damage) that is contingent on a saving throw, the damage of that second source is a different damage roll from the first.
@RubiksMoose how do you distinguish between second damage roll and extra damage?
@JeremyECrawford Separate damage rolls are delivered by separate attack rolls, saving throws, or other processes. Extra/bonus/additional damage is called extra/bonus/additional damage.

Both of these effects add extra dice, where Fueled Spellfire calculates the total before adding to the damage roll. That's how I'd read it anyway, it's similar enough that I could see it going the other way (and I don't think the extra damage is going to be a huge deal, to be honest, it's only up to 2d12 at most if you're multiclassed in Barbarian and choosing to cast a spell). Plus it consumes a resource.

It makes sense. I am not completely convinced the wording was done on purpose though :D Would there be any way to ask the devs directly?

@quadhund If I make 1 attack with a longsword as a part of casting Green Flame Blade, while having the Hex spell affecting the target, and then I use Divine Smite once the attack hits, how many damage rolls are there?

@JeremyECrawford When something in the game (Sneak Attack, Divine Smite, hex, etc.) causes your attack/spell/etc. to deal extra dice of damage, those dice are added to the damage the effect is already dealing, if any. It's one big damage roll, extra damage included. 
@DaveWil33 Would only be a second damage roll if there was something locked behind a save like poison damage, yes?
@JeremyECrawford If an attack has a damage roll but also a second damage roll (not extra damage) that is contingent on a saving throw, the damage of that second source is a different damage roll from the first.
@RubiksMoose how do you distinguish between second damage roll and extra damage?
@JeremyECrawford Separate damage rolls are delivered by separate attack rolls, saving throws, or other processes. Extra/bonus/additional damage is called extra/bonus/additional damage.

If I interpret it correctly, that would mean that the spellfire damage is doubled with a crit. 

Oh no, The confusion intensifies :D

Well, if you consider it's a second damage roll that is contingent on resolving a Saving Throw or a spell attack to determine whether you're dealing Radiant damage, the damage from this second source (Fueled Spellfire) would be a different damage roll from the first one.

Quote from Jayce_Danathar >>

Both of these effects add extra dice, where Fueled Spellfire calculates the total before adding to the damage roll. That's how I'd read it anyway, it's similar enough that I could see it going the other way (and I don't think the extra damage is going to be a huge deal, to be honest, it's only up to 2d12 at most if you're multiclassed in Barbarian and choosing to cast a spell). Plus it consumes a resource.

It makes sense. I am not completely convinced the wording was done on purpose though :D Would there be any way to ask the devs directly?

@quadhund If I make 1 attack with a longsword as a part of casting Green Flame Blade, while having the Hex spell affecting the target, and then I use Divine Smite once the attack hits, how many damage rolls are there?

@JeremyECrawford When something in the game (Sneak Attack, Divine Smite, hex, etc.) causes your attack/spell/etc. to deal extra dice of damage, those dice are added to the damage the effect is already dealing, if any. It's one big damage roll, extra damage included. 
@DaveWil33 Would only be a second damage roll if there was something locked behind a save like poison damage, yes?
@JeremyECrawford If an attack has a damage roll but also a second damage roll (not extra damage) that is contingent on a saving throw, the damage of that second source is a different damage roll from the first.
@RubiksMoose how do you distinguish between second damage roll and extra damage?
@JeremyECrawford Separate damage rolls are delivered by separate attack rolls, saving throws, or other processes. Extra/bonus/additional damage is called extra/bonus/additional damage.

If I interpret it correctly, that would mean that the spellfire damage is doubled with a crit. 

Oh no, The confusion intensifies :D

No, because you aren't adding the dice to the damage of attack. You are adding the result of the dice to the damage of the attack.