As you stated in Dueling (PHB, 72)
When you are wielding a melee weapon in one hand and no other weapons
And that Alchemist's Fire is listed under Adventuring Gear (PHB, 150), but it is clearly intended to be an improvised weapon, per the book as well (PHB, 150-151)
...treating the alchemist's fire as an improvised weapon
And Improvised Weapons are under the Weapons list from PHB, 147.
An improvised weapon includes any object you can wield in one or two hands, such as broken glass, a table leg, a frying pan, a wagon wheel, or a dead goblin.
Other items under Adventuring Gear are things like ammunition and I think it's clear that for those and for Alchemist Fire, Acid, etc. that they are improvised weapons when being used. Which is now why I will discuss the difference between holding an object in your hand and wielding an improvised weapon in your hand.
The difference is that the alchemical fire is basically a grenade. It's an object until activated (thrown) but the intent is that it's a weapon.
Holding vs Wielding
If you're just holding with no intent to use, the RAW would likely point to allowing the Dueling style as you aren't planning on using what you're holding as a weapon.
However, if you intend to use the object in your hand as a weapon, then you would no longer be Dueling.
While a DM could allow this based on the looseness of the rules, they are ignoring the spirit of the Dueling Fighting Style: which is that you are wielding a one-handed weapon and nothing else. If your offhand is holding something that you intend to use as a weapon, then you are no longer Dueling.
In addition, there isn't a RAW way to get an offhand attack unless they use an Action Surge which limits how often this could be done or the instances in which it may be a concern. It still is the equivalent of having another weapon 'ready' if something were to happen to the primary weapon.
You can try and do a Deception check against your DM to convince them that the item you're holding isn't intended to be an improvised weapon, but that DC may be pretty high.
That being said, as long as you don't do both on your turn then I don't think it affects overall balance much. You're only removing the free action(?) to pull the flask out before throwing it. Not a huge deal, just seems more "duelist" to not do it. Unless it's one of those grenade duels.