Elytra texture transparencies become opaque if any of the following are true:
player is holding an item in the main or off hand
player is wearing a pumpkin
player is wearing a player head (Not affected by creeper, skeleton, wither skeleton, zombie, or dragon heads)
player is Wearing an armor piece with an enchantment glow (If the elytra has an enchantment glow, it works properly)
Wearing the elytra without meeting any of the conditions listed above will render the texture as expected, with all transparencies working. I've included two comparison screenshots to demonstrate this, and the texture file used so you can test it for yourself. These textures were done via a resource pack.
I included a screenshot of the debug screen in case I missed anything in the environment. Yes, the screenshot is in 1.9-pre4, but I also just tested it in 1.9 (released as I'm typing this), and can confirm that the bug is still there.
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Transparency is not fully supported on entity textures, however this was recently fixed for the main player skin so maybe this is valid.

Once it's taken care of, it'll allow for more customization too. The fact that it's part of the player is pretty much the only reason I posted this. Besides, it's kinda working in the first place.

More than likely, Grum or Prof needs to add a call to GLStateManager.enableBlend(); before rendering ElytraLayer. Just guessing on the name here.
From what I recall, ItemRenderer calls GLStateManager.disableBlend() on its way out of renderStatic, which accounts for the issue being present in the first three listed cases. In fourth case, rendering the foil layer of armor probably disables blending on its way out.