The Tortle's Natural Armor ability says:
Due to your shell and the shape of your body, you are ill-suited to wearing armor. Your shell provides ample protection, however; it gives you a base AC of 17 (your Dexterity modifier doesn’t affect this number). You gain no benefit from wearing armor, but if you are using a shield, you can apply the shield’s bonus as normal.
The Monk's Unarmored Movement ability says:
Starting at 2nd level, your speed increases by 10 feet while you are not wearing armor or wielding a shield. This bonus increases when you reach certain monk levels, as shown in the Monk table.
The comments on this answer to a related question briefly mention the interaction between Natural Armor and Unarmored Movement, but nothing conclusive is discussed.
It is not clear if Natural Armor counts as worn, or even as armor, for the purposes of the Monk's Unarmored Movement Ability.
Does a Tortle's Natural Armor interfere with the Monk's Unarmored Movement ability?