Leashes are not occluded correctly, eg. they aren't visible when the animal which they are connected to isn't visible.
To reproduce:
Attach a leash to an animal and then to a fence(important, it works in hand for me).
Put your view that you see the fence but not the animal. And then leash is not rendered. Of course it would be even harder to handle it when both animal and fence are not visible, but it should work correctly when at least one of them is.
Linked issues
Comments 2
This ticket has been marked as a duplicate of MC-14415, in favor of the better description/information provided.
I have doubts that this will correct the issue, but OpenJDK 1.6 has been known to occasionally cause issues with Minecraft, and I would recommend an upgrade to either Oracle java and/or java 1.7.