No. Neither Shields nor Improvised Weapons are actually Weapons.
(They can be treated as weapons during a melee attack with them, but not otherwise. Some improvised weapons are actually still weapons, but leave that to the other question I asked.)
There is a relevant tweet from Jeremy Crawford:
Q: A shield as an improvised weapon with the Dual Wielding feat: Does the feat's +1 AC stack with the shield +2 AC that round?
J.C.: Dual Wielder is meant to work (RAI) with a melee weapon or an equivalent, not something like a shield.
Dual Wielder (PHB p165) requires the character to be wielding a melee weapon in each hand.
Common weapons are listed in the table in the PHB p149. While you may still make some weapon attacks with items not found in this table, it is not a weapon unless it is on the table or explicitly defined as such.
An example of an improvised weapon resembling an actual weapon is the leg of a chair. It resembles a club, and can be treated as one. A Shield does not resemble any actual weapon. Likewise, "frying pans, wagon wheels, and dead goblins" do not become weapons because you attack with them. (Those are some of the examples in the PHB.)
Apparently, during the actual attack, an improvised weapon is temporarily considered a weapon. However, this temporary status won't qualify it as a weapon except for features which apply to that attack. So Booming Blade would work with an improvised weapon but Pact of the Blade would not.
- Unarmed Strikes
- unarmed strikes are classified as melee weapon attacks, but they (fists, legs, etc.) are not actually weapons (this is in the PHB errata)
- Shield Shove
- granted by the Shield Master feat (PHB p170) as a bonus action
- Improvised Weapons (PHB p147)
- Ranged Weapons and Ammunition
- since these are not melee weapons, they can only be used for melee attacks as improvised weapons (and might be treated as a simple melee weapon such as a club or dagger if similar enough)
- you ignore all of its properties (i.e., ranged, heavy, light, etc.)
In order to qualify for the AC bonus granted by Dual Wielder, you must be wielding 2 actual distinct weapons, 1 in each hand. This means you can't use unarmed strikes, shields, improvised weapons, double weapons, etc., unless the item itself has a specific rule that applies. For instance, I can imagine Spiked Chain or some other exotic double weapons being allowed to qualify for Dual Wielder if specifically called out in their description.
Tavern Brawler (PHB p170) does not make improvised weapons into actual weapons. All it does is allow proficiency when using them (plus the other parts of the feat).