Devil's Sight can see through Maddening Darkness, but doesn't protect you from the damage
Excerpted, Maddening Darkness has the following effects:
Magical darkness spreads from a point you choose ... Non-magical light, as well as light created by spells of 8th level or lower, can't illuminate the area.
... Whenever a creature starts its turn in the sphere, it must make a Wisdom saving throw, taking 8d8 psychic damage on a failed save, or half as much damage on a successful one.
As the first effect of Maddening Darkness is simply to produce magical darkness, that particular aspect does not affect a Warlock with Devil's Sight, who can see through magical darkness without issue.
However, the second effect does not depend on the darkness, it simply applies to the same area (it's in a separate paragraph from the description of the darkness and doesn't reference the darkness at all, either). Anyone starting their turn in the area takes the psychic damage that the spell deals - seeing through the darkness doesn't change that, and the spell has no exception which makes the caster immune. You could even illuminate the area using a light-causing spell cast at 9th level, but that would not prevent the damage.