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I am fairly certain the answer is no but I just wanted to confirm.
In the PHB p.20 dwarves have the feature:
Speed. Your base walking speed is 25 feel. Your speed is not reduced by wearing heavy armor.
Monks have a feature called Unarmored Movement that increases their movespeed while not wearing armor (PHB p.78)
Would the dwarf speed feature counteract the unarmored requirement for Monks Unarmored Movement?
PHB p.78 says:
Unarmored Movement
Starting at 2nd level, your speed increases by 10 feet while you are not wearing armor or wielding a shield. [...]
Rearranging that statement to make the logic explicit gives us:
If you are not wearing armor or wielding a shield
then your speed increases by 10 feet.
If the premise doesn't hold (i.e. you are wearing armor) then the conclusion (i.e. your speed increases) doesn't follow.
So, no, dwarven monks don't get any of the benefits of the monk's Unarmored Movement feature when they're wearing armour. Of course they're not slowed by the armor, they just don't get the speed increase or the other features of Unarmored Movement.
