Does Alchemist's Fire damage get rolled once, or every turn?

Question

Alchemist's Fire does things quite differently from other items, being an improvised weapon that must always use ranged attacks, and deals its damage on subsequent turns, not when thrown. And Jeremy Crawford has tweeted a number of clarifications about how the damage works. But there's one thing that I haven't found any answers about, whether official, tweet, or even just accepted common practice. Does the damage get re-rolled every turn, or only the once?

On the one hand, only once would make sense with Crawford's tweets; Dexterity gets added to the damage it deals, because that damage is the one roll made for the weapon's damage, and the usual rule applies. Which feels more sensible than adding Dex to every roll, turn after turn.

On the other hand, there's no rule to explicitly say that it's rolled only once, and ongoing damage is usually re-rolled every time.

So, is the idea supposed to be that the improvised range weapon deals 1d4+dex damage, and the target takes that same damage every turn, or is it that they take a re-rolled 1d4+dex every turn?

For context: I DM, and a player with a Fast Hands-using Thief just brought this up. The whole Dex-to-ongoing-damage thing had always felt a bit off to both of us. He had an idea that his might be the intended interpretation, and I can't actually find anything anywhere that actually definitively rules either way.

Answer 1

On your turn when you throw the Alchemist's Fire (PHB 148), it is treated as an improvised weapon and deals 1d4+Dex bludgeoning damage (or whichever type DM chooses; see PHB 148 for the rules for improvised weapons).

The Dex modifier is added to the use as an improvised weapon because the rules on damage rolls state (PHB 196):

When attacking with a weapon, you add your ability modifier — the same modifier used for the attack roll — to the damage.

The important distinction comes here in the description of Alchemist's Fire:

On a hit, the target takes 1d4 fire damage at the start of each of its turns.

The fire damage does not occur until the start of the target's turn, ergo the damage roll from using it as an improvised weapon is irrelevant now.

So at the start of the target's turn, you roll 1d4 fire damage, unless the target has expended an action to extinguish it on a previous turn. To be clear, the damage that occurs at the start of each turn is damage from Alchemist's Fire, not damage from improvised weapon, so it's certainly not going to add the Dex modifier.