14 There remain hereafter, in the due time of the Lord, other bishops to be set apart unto the church, to minister even according to the first;
15 Wherefore they shall be high priests who are worthy, and they shall be appointed by the First Presidency of the Melchizedek Priesthood, except they be literal descendants of Aaron.
16 And if they be literal descendants of Aaron they have a legal right to the bishopric, if they are the firstborn among the sons of Aaron;
17 For the firstborn holds the right of the presidency over this priesthood, and the keys or authority of the same.
18 No man has a legal right to this office, to hold the keys of this priesthood, except he be a literal descendant and the firstborn of Aaron.
19 But, as a high priest of the Melchizedek Priesthood has authority to officiate in all the lesser offices he may officiate in the office of bishop when no literal descendant of Aaron can be found, provided he is called and set apart and ordained unto this power, under the hands of the First Presidency of the Melchizedek Priesthood.
20 And a literal descendant of Aaron, also, must be designated by this Presidency, and found worthy, and anointed, and ordained under the hands of this Presidency, otherwise they are not legally authorized to officiate in their priesthood.
D&C 107:15-17, 71-76
15 The bishopric is the presidency of this priesthood, and holds the keys or authority of the same.
16 No man has a legal right to this office, to hold the keys of this priesthood, except he be a literal descendant of Aaron….
69 Nevertheless a bishop must be chosen from the High Priesthood, unless he is a literal descendant of Aaron;
70 For unless he is a literal descendant of Aaron he cannot hold the keys of that priesthood.
71 Nevertheless, a high priest, that is, after the order of Melchizedek, may be set apart unto the ministering of temporal things, having a knowledge of them by the Spirit of truth;
72 And also to be a judge in Israel, to do the business of the church, to sit in judgment upon transgressors upon testimony as it shall be laid before him according to the laws, by the assistance of his counselors, whom he has chosen or will choose among the elders of the church.
73 This is the duty of a bishop who is not a literal descendant of Aaron, but has been ordained to the High Priesthood after the order of Melchizedek.
74 Thus shall he be a judge, even a common judge among the inhabitants of Zion, or in a stake of Zion, or in any branch of the church where he shall be set apart unto this ministry, until the borders of Zion are enlarged and it becomes necessary to have other bishops or judges in Zion or elsewhere.
75 And inasmuch as there are other bishops appointed they shall act in the same office.
76 But a literal descendant of Aaron has a legal right to the presidency of this priesthood, to the keys of this ministry, to act in the office of bishop independently, without counselors, except in a case where a President of the High Priesthood, after the order of Melchizedek, is tried, to sit as a judge in Israel.
Joseph Fielding Smith (Quorum of the Twelve)
There are some men in the Church who have been blessed by patriarchs and pronounced descendants of Levi, but they have not made any claim to the office of bishop, for the revelation governing this situation says literal descendant of Aaron, not of Levi. There is evidently a great host of men who are descendants of Levi but not of Aaron.
The person spoken of in the revelations as having the right by lineage to the bishopric is the one who is the firstborn. By virtue of his birth he is entitled to hold “the keys or authority of the same.” This has reference only to the one who presides over the Aaronic Priesthood. It has no reference whatever to bishops of wards. Further, such a one must be designated by the First Presidency of the Church and receive his anointing and ordination under their hands. The revelation comes from the Presidency, not from the patriarch, to establish a claim to the right to preside in this office. In the absence of knowledge concerning such a descendant, any high priest, chosen by the Presidency, may hold the office of Presiding Bishop and serve with counselors. (Doctrines of Salvation, 3:92-93)
Bruce R. McConkie
From Aaron to the coming of John the Baptist, the high priests in Israel served in their presiding offices (of the Aaronic order) because they were descendants of Aaron. The office of Presiding Bishop in the Church today is of comparable hereditary nature, although the Lord has not so far designated the lineage in which the right to such office rests (D&C 68:14-24). The right to hold the Levitical Priesthood anciently was limited to the sons of Levi, who thus gained their priesthood prerogatives by birth. In the meridian of time our Lord altered this system and spread this Aaronic order of authority among worthy male members of the Church generally (1 Tim. 3:1-13). (Mormon Doctrine, p.87-88)
Harold B. Lee (Quorum of the Twelve)
I remember a man, a big, heavyset, rough-looking fellow, dressed in a working garb, came into my office, and after a few pleasantries and a little introduction, he said to me as though to startle me, “I’m a literal descendant of Aaron.” Well, I said, “I have always been curious to see a literal descendant of Aaron.” And when he saw that I wasn’t too much impressed, he said, “And I have come to claim my right to be the Presiding Bishop of the Church.” “Well, now,” I said, “that is very well, but there is just one little matter that you have overlooked.” And he wanted to know what that was. And then I read to him from the revelations [that] a literal descendant of Aaron may serve without counselors, if called by the President of the Church and ordained to that office (see D&C 68:20; 107:76). “Now you just go back home and wait until the President of the Church sends for you, and then if he ever does send for you, you won’t have to have counselors to be the Presiding Bishop of the Church.” Well, he had lost that vital thing, which, if we understand, would keep us from being led astray. He failed to understand that only through that one man who is His mouthpiece on earth will the Lord reveal instructions for His church. (Teachings of Harold B. Lee, p.545)