Yes Ezekiel, this also happens with the current version (1.6.2 SSP), at least to me. It happens when the chunk (the animal is tied on) unloads and loads again.
Example: I ride with my mule from my base to a mine, then I tie it to a post, enter the mine (which is huge, +400 blocks straight), mine some goodies and then, when I go back to the mule to unload the inventory, I find the animal untied and wandering a couple of blocks away and the leash item dropped on the ground next to the post. Gotta contribute with screenshots, though.
"So I tied a horse to a fence then closed out of Minecraft, then later when I came back on the leash was gone, they still acted like there was a leash BUT THE VISUAL LEASH WAS GONE"
@Ymus: IT IS NOT a duplicate, they are different bugs. If you read carefully, in MC-14826 it says that the leash VISUALLY disappears, but the animal is still tied to the fence. In this one, the leash BREAKS, it drops on the ground, and the animal is set loose.