The prohibitions of Shabbat apply not only to people, but also to their animals and pets. Maran in Siman 305 discusses "Shevitat Behemah"-the prohibition of having an animal owned by a Jew perform Melacha on Shabbat. For example, it is a Torah prohibition to allow one’s donkey to carry a load in the public domain. The Shulhan Aruch outlines what items are prohibited as burden and what is permitted as a protective accessory for the animal. Just like a person is allowed to wear clothing in the public domain, so too, an animal may also do so. For example, it is permitted for a donkey to wear a "Mard’at"-a type of sweater to keep it warm. On the other hand, an animal may not wear an ornament in the public domain. For an animal, it is considered carrying, whereas for people it is considered normal jewelry.
The Halacha permits walking a dog by holding its leash, if two conditions are met. First, no more than a Tefah (handbreadth=3-4 inches) is drooping from his hand. If more is hanging from the other side of his hand, it appears that he is carrying it. Second, the leash must always remain at least one Tefah above the ground. If the leash is too long, it may be wrapped around the neck of the dog.
The Aruch Ha’shulhan (Rav Yechiel Michel Epstein of Nevarduk, 1829-1908) strongly objects to allowing a dog to wear its tag in the public domain. He rules that it is not considered a garment, and even if it would be an ornament, it is not permitted. Accordingly, one may not take a dog for a walk on Shabbat in the public domain wearing such a tag. The Shemirat Shabbat K’hilhata (27:9) rules in accordance with this position. However, in note 33, he quotes Rav Shlomo Zalman Auerbach (Jerusalem, 1910-1995) who suggests a lenient approach, based on the fact that the tag is for the benefit of the animal to protect it from becoming impounded by the authorities.
However, the restriction regarding dog tags does not apply to Yom Tob. Maran rules that one is permitted to carry on Yom Tob, even not for the sake of the Hag. Although Hacham Bension rules that one should be stringent, nevertheless, in this case it would be permitted, because of the opinion of Rav Shlomo Zalman Auerbach, as well as the opinions that hold that "Shevitat Behemah" does not apply to Yom Tob.
One may walk a dog on Shabbat provided that the leash does not droop within 3 inches of the ground or more than three inches on the other side of his hand. In addition, the dog’s tags should be removed before walking on Shabbat, but on Yom Tob, they may remain.