Chests are also not rendered in water if GUI is disabled (by pressing F1). I experienced this bug myself, but was able to fix it by adding the textures to an Atlas. I'd recommend to use chest atlas and then render with that like vanilla does.

Abnormals Core: 3.0.6
Atmospheric: 3.0.0

With GUI and targetting:

Without GUI and targetting:

With GUI but not targetting:

I am unable to reproduce this.
Do you have any rendering mods like Optifine running?

@SmellyModder No, only Atmospheric and Abnormals Core. It can be reproduced simply by stepping back and pressing F1, unless graphics are set to Fabulous mode.

I got it to happen a few times now. The conditions for it to occur are pretty finicky.
For now, use Fabulous graphics to get around this issue.