Quoted from d20:
Base Speed: (Slow and Steady) Dwarves have a base speed of 20 feet,
but their speed is never modified by armor or encumbrance.
and
A barbarian’s land speed is faster than the norm for her race by +10
feet. This benefit applies only when he is wearing no armor, light
armor, or medium armor, and not carrying a heavy load. Apply this
bonus before modifying the barbarian’s speed because of any load
carried or armor worn. This bonus stacks with any other bonuses to the
barbarian’s land speed.
So a dwarf can never be slowed by any kind of armor, but barbarians lose the +10 feet of movement when wearing heavy armor.
Which takes priority over the other?
Best Answer
They do not conflict.
The dwarf's base speed is never modified by armor or encumbrance. Their base speed is 20.
The barbarian gains a bonus to base speed when not wearing heavy armor (and not carrying a heavy load). Once they wear heavy armor (or carry a heavy load), the class feature no longer has its conditions met, and thus does not provide the bonus, regardless of being a dwarf or not.
Therefore, as a dwarf barbarian, when wearing non-heavy armor and not carrying a heavy load, your base speed is 30, and is not affected by armor or encumbrance. When wearing heavy armor or carrying a heavy load, your base speed is 20, and is still not affected by encumbrance or armor.