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?
@RichardMassive intended - shields are a little too flexible to give out as easily.