One is that it could be considered unnecessary as in an ornament or a muzzle (the question being if a muzzle is necessary or not). A nose-ring on a camel, for example: some say “Whether as an ornament or a guard, it is forbidden” others “as an ornament it is forbidden, as a guard it is permitted.”
The second concern is that if the rein or chain slipped off the animal, the person holding it would then be left carrying an object on Shabbat!
But the Mishnah is specific: A HORSE WITH ITS CHAIN, AND ALL CHAINWEARING ANIMALS MAY GO OUT WITH THEIR CHAINS AND BE LED BY THEIR CHAINS.
But what is the difference between “going out” and “being led” ? Why does the Mishnah list both?
R. Huna suggests that these animals can “go out” wearing a chain which is loosely wrapped around them, ready to be used if necessary – or with someone holding the chain and leading them. Samuel disagrees saying they cannot wear the chain unless someone is holding it. Huna seems to win.
But the interesting side reference is to a “red heifer.” (Always interesting when the red heifer is mentioned.) How does it relate? It seems that this it is permitted to lead it or tie it with a chord. But the specific biblical command is it must be an animal “upon which never came a yoke” (Num. 19:2). So a cord or lead is not considered a “burden” making it permissible.
Tell that to the animal!