The priest who is chief among his brothers, on whose head the anointing oil is poured and who has been consecrated to wear the garments, shall not let the hair of his head hang loose nor tear his clothes.
Leviticus 21:10 ESV
The High Priest is not to uncover his head or tear his clothes in mourning. He is never to be out of character for any reason while he is on duty.
And the daughter of any priest, if she profanes herself by whoring, profanes her father; she shall be burned with fire.
Leviticus 21:9 ESV
This is about Temple prostitution. Like Nadab and Abihu, the priest's daughter offers "strange fire", making herself "set apart" in the process. Like other objects that are illicitly made holy, she is to be devoted to destruction, i.e. burned. Jewish tradition says she is to be stoned first and then her body burned, but the simple reading implies that she is to be burned to death.
Here's a list of Apostolic passages to read and study with parsha Emor ("speak" Leviticus 21:1-24:23), plus related commentary and videos: https://www.americantorah.com/....2021/02/23/parsha-em