Leviticus 21:12

And he shall not go forth from the sanctuary, or profane the sanctuary of his God; for the anointing oil of his God is upon him: I am the Lord.

He must not leave the Sanctuary The Exegetes correctly interpreted this verse [Sanhedrin 18a] to mean: He will not leave the Sanctuary to follow the funeral procession. Some people interpret the verse literally, and apply it to the seven days of consecration; but this is implausible, because Scripture explicitly discusses the kohen who already “has been consecrated” [:10].

NEITHER SHALL HE GO OUT OF THE SANCTUARY. “He does not follow the bier [of a near relative]. Our Rabbis have further derived from this verse [the principle] that the High Priest may perform [the Divine Service] when he is an onen (a mourner),, Note 85. the meaning of the verse thus being as follows: Even if his father or his mother died, he need not leave the Sanctuary but he may perform the Service. Nor shall he profane the Sanctuary of his G-d, means that [the High Priest] does not profane thereby the Service by ministering when he is an onen,, Note 85. since Scripture permitted him to minister. From this you deduce that a common priest who officiated when he is an onen, does profane the Service.” This is Rashi’s language.
But these two interpretations [that Rashi mentioned, namely, that the High Priest is not to follow the bier of a near relative, and that he may officiate when he is an onen], are really opposing opinions, for since the verse permitted the High Priest not to leave the Sanctuary on account of his mourning, as a common priest must do, but instead he may [continue to] officiate when he is an onen,, Note 85. then there is no admonition here against his following the bier of his near relative.. In interpreting the phrase, neither shall he go out of the Sanctuary as a prohibition, meaning that he is not permitted to leave the Sanctuary to follow the bier, how could Rashi continue on the basis of the same phrase to state that he need not leave the Sanctuary, but he may perform the Service, since these two interpretations of the same phrase are contradictory to each other! This is the meaning of Ramban’s succinct objection: “Since the verse permitted the High Priest … then there is no admonition …” Mizrachi, in defense of Rashi, answers that Rashi was not referring only to the first phrase [as if both the permission and the admonition are derived from the same phrase], but rather to both phrases of the verse, the first one [neither shall he go out …] establishing the prohibition, and the second one [nor profane …] constituting the negation. Ramban will also further on defend Rashi’s position from another standpoint. Similarly, nor shall he profane the Sanctuary of his G-d is, according to this interpretation [of Rashi], nothing but a statement that the Service of the High Priest does not become profaned …

ומן המקדש לא יצא. According to Rabbi Saadyah gaon, this means primarily joining the funeral cortege which requires leaving the holy precincts.

ולא יחלל את מקדש אלוקיו, so as not to desecrate the Temple of his G’d, as people would conclude that giving honour to the dead is more important than to remain within the sphere of G’d’s immediate presence, something he enjoys as the exceptional privilege of being exposed to on an ongoing basis.