Okay, so I've been playing on a private server with a few friends, and we have the Cyberware mod installed. Everything has been working great up until we started playing around with adding Cyberlimbs onto our characters.
You see, we also have a mod by the name of "Cosmetic Armor: Reworked", which is another mod that we love and enjoy. We really don't want to remove either of them, but there's been a bit of a bug which is causing some issues.

Here's a screenshot of my regular and cosmetic armor slots before applying any cyberlimbs
http://i.imgur.com/skQushK.png

All seems fine, right? But sadly, here's what happens as soon as I add any limbs on.
http://i.imgur.com/JxGjq9O.png

It completely breaks the normal armor menu, showing my cosmetic piece as actual armor. When I try to click and drag a piece off, this happens:
http://i.imgur.com/PV6qGwt.png

The armor and elytra are still there, but for some reason, whenever the limb is applied I cannot use the elytra and any functions the armor pieces would have no longer work. With modded armor that adds abilities, those abilities no longer work. As for a vanilla example, I can no longer glide with the elytra unless you toggle and remove all cosmetic armor pieces. Now, this is more likely an issue with the Cosmetic Armor mod, but we'd love to keep both mods.
If you have any means of fixing this or giving advice, it'd be greatly appreciated.

This also happens with WearableBackpacks
And i assume anything which handles in a similar way

For now you can try disabling the rendering of Cyberlimbs in the config file - I can try to find a more substantial fix down the line

I'm having a similar issue, but it's with trinkets and robes from Botania, and Angel wings from Extra Ultilities 2. And it only seams to start after I put on Synthetic skin on to hide my limbs. But they don't reappear if I take off the skin.