The following entity parts are supposed to still be visible when the entity is under the invisibility potion/status effect. (Based on the Java edition behavior). However, they are completely invisible in MCPE.
horse armor
pig saddles
strider saddles
llama carpets
spider eyes (fixed, but should be partially transparent)
enderman eyes (fixed, but should be partially transparent)
In the newest version no entity overlays are all visible this includes all saddle types but it doesn't include ender and spider eyes thier still visible also the invisiblility doesn't effect shulkers they don't turn invisible at all
Confirmed on Samsung Galaxy Note 5. Android 5.1.1 Version: 0.12.2