Does a wielded shield count as an off-hand weapon for purposes of dueling style? My usual image of dueling is a fencer, which certainly doesn't use a shield, but I'm not sure of the actual intent of the rule.

No, a Shield does not count as an off-hand weapon (or any weapon). You can absolutely use a 1H weapon & Shield while getting the bonus from the Dueling Fighting Style.

You could start wielding a shield as an offhand weapon by making an improvised attack with it, but it's unclear how long that toggle lasts because that scenario wasn't explored in the rules text (just while making the attack? for the rest of your turn? for the rest of the round until the start of your next turn?). In general, wielding a shield does not prevent you from applying the Dueling fighting style bonus to your other weapon though, and your DM is unlikely to ever raise an issue unless you start trying to use Dueling and Two Weapon Fighting bonus attacks on the same round with fancy interaction orders. 

Quote from Chicken_Champ >>

You could start wielding a shield as an offhand weapon by making an improvised attack with it, but it's unclear how long that toggle lasts because that scenario wasn't explored in the rules text (just while making the attack? for the rest of your turn? for the rest of the round until the start of your next turn?). In general, wielding a shield does not prevent you from applying the Dueling fighting style bonus to your other weapon though, and your DM is unlikely to ever raise an issue unless you start trying to use Dueling and Two Weapon Fighting bonus attacks on the same round with fancy interaction orders. 

Now that's a munchkin move.... :)

Two Weapon Fighting requires the use of light weapons. A shield as an improvised weapon is not light, so no two attacks.

As for the original question, yes, using a shield does not prevent getting the dueling fighting style benefit.

Let's not go there. A shield is not a weapon. It does not ever qualify for two-weapon fighting. Using one as an improvised weapon is already completely in the domain of house rules/DM fiat. I believe the OP wants to know how the interaction does work in 5e, not how it could work with speculative house rules.

Not with Dual Wielder feat it doesn't. It's an angle, for a character that wants the Shield's +2 AC bonus and cares more about making a Bonus Attack than the base damage die of that attack. 

Its not speculative. Anything that can be held in one or two hands can be used as an improvised weapon, RAW, and any improvised weapon which doesn't resemble a weapon does 1d4 and requires Improvised Weapon Proficiency  (see Tavern Brawler) in order to add your Proficiency Bonus to the attack. None of that is optional or dependent on the DM's fiat, it's printed RAW rules. 

I spit in my hand, and throw it at you for 1d4 damage. It's DM fiat. The RAW is literally that it is always DM fiat.

Calling the moisture in your hand an object would certainly require DM fiat and consent. 

"Literally," the rule is:

Sometimes characters don't have their weapons and have to attack with whatever is at hand. 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.

Often, an improvised weapon is similar to an actual weapon and can be treated as such. For example, a table leg is akin to a club. At the DM's option, a character proficient with a weapon can use a similar object as if it were that weapon and use his or her proficiency bonus.

An object that bears no resemblance to a weapon deals 1d4 damage (the DM assigns a damage type appropriate to the object). If a character uses a ranged weapon to make a melee attack, or throws a melee weapon that does not have the thrown property, it also deals 1d4 damage. An improvised thrown weapon has a normal range of 20 feet and a long range of 60 feet.

Quote from Chicken_Champ >>

Not with Dual Wielder feat it doesn't. It's an angle, for a character that wants the Shield's +2 AC bonus and cares more about making a Bonus Attack than the base damage die of that attack. 

Its not speculative. Anything that can be held in one or two hands can be used as an improvised weapon, RAW, and any improvised weapon which doesn't resemble a weapon does 1d4 and requires Improvised Weapon Proficiency  (see Tavern Brawler) in order to add your Proficiency Bonus to the attack. None of that is optional or dependent on the DM's fiat, it's printed RAW rules. 

And as a DM I wouldn't have an issue with that as the player has obviously invested into getting that combo, so let them use it. But then it's a weapon and the dueling fighting style doesn't apply.

Assuming a character has Dueling, Two Weapon, and Dual Wielder Feat, they can:

• Two weapon fight with main weapon and improvised weapon (shield) and apply attack damage bonus to improvised weapon.

OR

• Dueling fight with main weapon and get +2 bonus damage.

Each is a different attack action. The two weapon fighting one seems better, but is the +2 AC worth giving up on wielding a better offhand weapon given that you have Dual Wielder?

All this is why I don't allow Feats for my players. They allowed silly things like this in 3e and should have been left there; I'm glad they're optional.

Harbinger, but how far in advance must that decision be made by the player? 

Player is squaring off against a foe with longsword and shield in hand, and Dueling fighting style, and Dual Wielder feat. They tell you, "I attack with my longsword" but don't tell you what they plan to do for the rest of the round (maybe they haven't even decided yet, this attack could change everything!). Do you tell them "if you apply your +2 dueling bonus to the damage of this attack, I'm not going to let you make any bonus action Improvised Weapon attacks with that shield"? If so, do they have to decide that before rolling the attack? Or after confirming the attack hits, but before rolling damage?

What about if they aren't wielding a shield, but instead just have a second sheathed longsword that they might want to draw and attack with later... do you tell them "if you apply your +2 dueling bonus to the damage of this attack, I'm not going to let you draw that other longsword and make a bonus attack with it"? Would you treat this situation differently than if say they were using their first attack to make a Versatile Longsword 2H attack (1d10), and then decide to draw a second longsword to make a bonus action attack (1d8)?

Either way, those sound like very reasonable house rules to discourage these shennanigans, but it would be a house rule, because the Dueling Fighting style in no way claims to prevent a character from using a bonus action to attack with another weapon. 

Quote from Chicken_Champ >>

Harbinger, but how far in advance must that decision be made by the player? 

Player is squaring off against a foe with longsword and shield in hand, and Dueling fighting style, and Dual Wielder feat. They tell you, "I attack with my longsword" but don't tell you what they plan to do for the rest of the round (maybe they haven't even decided yet, this attack could change everything!). Do you tell them "if you apply your +2 dueling bonus to the damage of this attack, I'm not going to let you make any bonus action Improvised Weapon attacks with that shield"? If so, do they have to decide that before rolling the attack? Or after confirming the attack hits, but before rolling damage?

What about if they aren't wielding a shield, but instead just have a second sheathed longsword that they might want to draw and attack with later... do you tell them "if you apply your +2 dueling bonus to the damage of this attack, I'm not going to let you draw that other longsword and make a bonus attack with it"? Would you treat this situation differently than if say they were using their first attack to make a Versatile Longsword 2H attack (1d10), and then decide to draw a second longsword to make a bonus action attack (1d8)?

Either way, those sound like very reasonable house rules to discourage these shennanigans, but it would be a house rule, because the Dueling Fighting style in no way claims to prevent a character from using a bonus action to attack with another weapon. 

Two weapon fighting specifies that you get the bonus action only if during your attack action you're holding a weapon in the other hand. You have to have drawn it at the latest in tandem with your normal action. This then means dueling can't be applied as you're holding two weapons.

When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use a bonus action to attack with a different light melee weapon that you're holding in the other hand.

True... unless we're talking about a 5th+ level martial character trying to apply the bonus to their first attack in the round, leaving open the possibility that they may still draw weapons before their second. You can't slip out of this quandary that easily! :D 

Regardless of how a DM chooses to rule on shields as a weapon, it is irrelevant to the original post.

Does a wielded shield count as an off-hand weapon for purposes of dueling style? My usual image of dueling is a fencer, which certainly doesn't use a shield, but I'm not sure of the actual intent of the rule.

The discussion of if (when) a shield in one hand prevents adding the Dueling fighting style bonus to an attack with the other hand has always been the point of this conversation, and every single post that has been made thus far has addressed that conversation. What are you getting at? 

Quote from Chicken_Champ >>

Does a wielded shield count as an off-hand weapon for purposes of dueling style? My usual image of dueling is a fencer, which certainly doesn't use a shield, but I'm not sure of the actual intent of the rule. 

The discussion of if (when) a shield in one hand prevents adding the Dueling fighting style bonus to an attack with the other hand has always been the point of this conversation, and every single post that has been made thus far has addressed that conversation. What are you getting at? 

 And the answer to the question is "no." A shield worn as a shield is not a weapon. If the question was about a shield brandished in your hands as a blunt instrument, it would have been a different question.

Making an improvised weapon attack with a shield does not cause it to stop functioning as a shield (+2 AC). 

By RAW, by RAI, and by historical precedent, a weapon in one hand and a shield in the other still works with a dueling fighting style. You are good to go.

So I can use two shields at once for +4 AC and then still attack with one of my improvised shields? :D

Shields. A shield is made from wood or metal and is carried in one hand. Wielding a shield increases your Armor Class by 2. You can benefit from only one shield at a time.

hehe you can use two shields at once, but as Chicken_Champ pointed out, you get no further benefit from it.