You can't put elytra on armor stands even though dispensers do place elytra on armor stands.
Code analysis by @unknown can be found in this comment
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is duplicated by 5
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The following is based on decompiled version of Minecraft 1.9 using MCP 9.24 beta. All method and class names are the names used in the decompiled version.
The reason why this happens is because the method net.minecraft.entity.item.EntityArmorStand.func_184199_a(EntityPlayer, Vec3d, ItemStack, EnumHand)
only tests if the item is a an instance of the ItemArmor
class (which is not the case) or if it is a skull or a pumpkin. It might be in general better to define for every item the slots it can be put in instead of having exceptions for all kind of items every time this information is needed.
The elytra is however not visible due to MC-89921
Dupe of MC-89921