The bug
Armor trim textures can be animated like any other texture which is part of an atlas can be (although each material must be specified manually due to MC-268543). However the base textures for armor themselves reject animated textures.
How to reproduce
A resource pack that assigns an animated texture to the diamond coast armor trim and to the iron armor texture is attached to this ticket.
Download and apply this resource pack
Look at something with the diamond coast trim on it
Look at an iron chestplate
Expected results
Both would look like an animated checkerboard.
Actual results
The diamond coast trim correctly looks like an animated checkerboard. However, the iron chestplate is not animated and incorrectly looks squished.
How to fix
The fix should be very simple - just add the armor textures to the armor_trims.png-atlas file, and make the game reference that for armor textures instead of the textures directly. The atlas name could possibly be changed to armor.png-atlas to reflect this change.
Can confirm this is still an issue on the latest release.