You gain no benefit from wearing armor.
This seems pretty unambiguous. There are no secret rules. Gaining the benefits of magic armor requires wearing it:
Unless an armor’s description says otherwise, armor must be worn for its magic to function.
But the Tortle’s Natural Armor feature says you gain no benefit from wearing armor. So usually, the magic of the armor is explicitly going to be a benefit of wearing it, unless the armor’s description says that it doesn’t have to be worn to gain magical benefits. (I’m not aware of magic armor that doesn’t require wearing to gain its magical benefits). We will see that while this is probably the RAW ruling, it really isn’t a big deal to rule either way.
It’s easy to work with this as a DM.
There are lots of magic items. I doubt you’re running a “only magic items are armor” campaign, so it should be easy enough to make sure your Tortle player has access to usable magic items. Maybe a particularly industrious magical craftsman specializes in doing armor enchantment transfers for Tortles.
There is some rules support for just letting it work.
We also see this in the rules for Wearing and Wielding Items:
In most cases, a magic item that's meant to be worn can fit a creature regardless of size or build. Many magic garments are made to be easily adjustable, or they magically adjust themselves to the wearer. Rare exceptions exist.
It’s open to interpretation, I think, if this paragraph carved out an exception to the Tortle’s Natural Armor feature, but it definitely would not be a big deal to let the Tortle benefit from magical armor, citing this rule as support.
With the Release of Monsters of the Multiverse, Tortles cannot even wear armor.
Tortles were revised for the publication of Modenkainen Presents: Monsters of the Multiverse, and their Natural Armor trait now reads:
You can’t wear light, medium, or heavy armor
So under the revised version, the question is moot.