Siman 523
Presented here for study, not as a practical ruling. The Shulchan Aruch rules by Sephardi custom, the Rema's glosses by Ashkenazi custom, and the Kitzur follows Ashkenazi custom. For a practical question, ask your rabbi.
א מְסִירִים זְבוּבִים הַנִּתְלִים בַּבְּהֵמָה, אַף עַל פִּי שֶׁהֵן עוֹשִׂין חַבּוּרָה.
1 Laws regarding animals on the holiday, 4 Seifim: We remove flies that are stuck on an animal, even though this creates a wound.
ב אֵין מְגָרְדִין בְּהֵמָה בְּמִגְרֶדֶת בְּיוֹם טוֹב, מִפְּנֵי שֶׁמַּשֶּׁרֶת שֵׂעָר.
2 We do not comb an animal with a fine comb on a holiday because this removes hair.
ג אֵין מְיַלְּדִין בְּהֵמָה בְּיוֹם טוֹב, אֲבָל מְסַעֲדִין אוֹתָהּ, שֶׁאוֹחֵז בַּוָּלָד שֶׁלֹּא יִפֹּל לָאָרֶץ, וְנוֹפֵחַ לוֹ בְּחוֹטְמוֹ, וְנוֹתֵן לוֹ דַּד לְתוֹךְ פִּיו.
3 We do not deliver an animal on a holiday, but we assist it by holding the newborn such that it does not fall on the ground, blowing in its nose, and putting the teat into its mouth.
ד בְּהֵמָה שֶׁרִחֲקָה וְלָדָהּ, מֻתָּר לְזַלֵּף (פי' לִשְׁפֹּךְ) מֵי שִׁלְיָתָהּ עָלָיו וְלִתֵּן מְלֹא אֶגְרוֹף מֶלַח בְּרַחְמָהּ, כְּדֵי שֶׁתְּרַחֵם עָלָיו; אֲבָל הַטְּמֵאָה אָסוּר לַעֲשׂוֹת לָהּ כֵּן. הַגָּה: וְעַיֵּן לְעֵיל סי' רמ"ו (ס"ג בְּהַגָּה) אִם אָדָם מְצֻוֶּה עַל שְׁבִיתַת בְּהֶמְתּוֹ בְּיוֹם טוֹב.
4 If an animal spurns its newborn, it is permitted to sprinkle (meaning to pour) the water of the after-birth on it and put salt in her womb so that she will have mercy on it. It is forbidden to do this for an impure animal. Rema: (And see above chapter 246 paragraph 3 in the Rem"a's note regarding whether a man is commanded to let his animal rest on a holiday).