According to D&D 5e designer Mike Mearls, the Mountain Dwarf is only proficient with light and medium armor and not proficient with shields.1
In RAW, if you want shield proficiency, you either need to pick up a level in a class with the proficiency or take the (optional) Moderately Armored feat, as no other feat grants shield proficiency.
I would highlight Mountain Dwarves are the only (sub)race that get any kind of armor proficiency. However, this is not totally redundant since the Mountain Dwarf subrace needs only one feat to gain shield proficiency (Moderately Armored) while all other character races need two (Lightly Armored and Moderately Armored). The designers may have felt two full feats, in addition to the other (sub)racial benefits, to be too much for the Mountain Dwarf,1 although it's somewhat ironic that so-called "shield dwarves" in Faerûn are not racially proficient with shields.
DMs may also house rule that other armor-related feats, such as Medium Armor Master, Heavily Armored or Shield Master, grant shield proficiency. In another apparent contradiction, the Shield Master feat (PHB, p. 170) neither prerequires nor grants proficiency with shields.
In a tweet exchange on November 2, 2014, @RichardMassive asked:
@mikemearls in the PHB things that give medium armor prof also give shields, with one exception: the Mountain Dwarf. oversight or intended?
Mearls responded:
@RichardMassive intended - shields are a little too flexible to give out as easily.
Composite Plating and Heavy Plating are intended to count as armor.
Keith Baker, original creator of Eberron and lead designer for the Wayfinder's Guide to Eberron, addresses this question (and many others) in the WGtE FAQ on his website:
Does the Warforged Heavy Plating Integrated Protection work with the Fighter’s Defense Fighting Style? Unclear on if (armor) means you’re “wearing armor” or not.
The intent is while you are using darkwood core you are not considered to be wearing armor for purposes of game effects such as Unarmored Defense, while you are considered to be wearing armor when you’re using one of the plating modes. However, as written Integrated Protection doesn’t allow you to benefit from the Defense Fighting Style. This won’t be changed until people have a chance to provide feedback on the race mechanics as they stand.
Baker points out that the feature doesn't specifically clarify that Composite Plating or Heavy Plating count as armor - but the parenthetical after each one reflects the intent, which is that Darkwood Core counts as being unarmored and that the other two options count as being armored. This is relevant for features like the Defense fighting style (which requires you to be armored), or the monk's Unarmored Movement feature (which, as the name implies, requires that you be unarmored).
In another response, he also says:
When a warforged is using the darkwood core (unarmored) mode and have light armor proficiency, can they choose whether they are considered to be armored or unarmored?
No. You are always considered to be unarmored when using darkwood core, and always considered to be armored when using plating.
So you can choose which of the three modes to use (depending on what armor you're proficient in), but you can't choose whether that mode counts as being armored.
...but as currently written, they don't count specifically as medium or heavy armor.
Baker also answers a related question about what kind of armor the Integrated Protection feature counts as:
Composite Plating and Heavy Plating both say “Armored” but do they count as medium and heavy armor respectively? Specifically does a warforged Barbarian in Heavy Plating mode lack access to the benefits of rage?
As written, the current version simply specifies that plating counts as armor, not that it counts specifically as heavy or medium armor. So in this version, a warforged barbarian can rage while in the heavy plating mode… though they will have to acquire the heavy armor proficiency before they can use heavy plating.
This may change in the future, but as of right now, a warforged barbarian can gain the benefits of Rage regardless of what defensive mode they're in. However, they can't benefit from features that forbid any armor at all while using Composite Plating or Heavy Plating.
Best Answer
Wearing non-proficient armor or shield incurs a penalty to attack rolls, STR ability checks, and DEX ability checks.
Each armor or shield has an Armor Check Penalty (or ACP), which is at least 0. The total ACP of your worn equipment is usually applied as a penalty to STR-based and DEX-based skill checks, such as Climb or Stealth.
If you aren't proficient with the armor you're wearing, then the ACP is also applied to your attack rolls (including combat maneuvers), STR ability checks, and DEX ability checks (including your initiative rolls). According to the core rules:
Note that there are exceptions, such as abilities that reduce an item's ACP. If the total ACP is 0, then these drawbacks are effectively negated, although arcane casters may still suffer spell failure chance.
(thanks to KRyan and Ifusaso for the clarification)