Commenting on the Davidic Kings as being “priests,” Old Testament scholar William Riley wrote the following:
The Cultic Role of David
The sacral character of the ancient Near Eastern monarch manifested itself in the priestly function of the king. As Keel observes:
Throughout the entire ancient Near East, but especially in ancient Sumer, cultic responsibilities devolve upon the king. The ancient Sumerian Ensi was as much priest as prince. He resides in the temple and is responsible for the welfare of the city god . . . As late as the Neo-Sumerian period, Ur-Nammu appears not only as temple builder but also in a priestly capacity. Iconographic evidence for the priestly functions of Mesopotamian kings are extant well into the latest Assyrian epoch. (O. Keel, The Symbolism of the Biblical World, pp. 278-79)
The reference to king as כהן in Ps. 110.4 testifies to the survival of this aspect of kingship in Israel, even though the priestly nature of Israelite kingship remain ambiguous in some Old Testament texts. (William Riley, King and Cultus in Chronicles: Worship and the Reinterpretation of History [JSOT 160; Sheffield: Journal for the Study of the Old Testament Press, 1993], 58)
In a footnote commenting on certain texts that seem, at first reading, to be ambiguous as to whether it is proper for a Davidic king to also serve as a priest, Riley noted:
Indeed, passages such as 1 Sam. 13.8-14 and 2 Chron. 26:16-20 seem to condemn kings who take it upon themselves to exercise a priestly function. However, the biblical author may not have placed the emphasis in 1 Sam. 13.8-14 on the cultic action as it appears at first place. Cultic activity was not incompatible with military or tribal leadership in pre-monarchial Israel, as can be seen from Jud. 6.24-26 or from the figure of Samuel himself. H.W. Hertzberg, I & II Samuel, p. 106, interprets the sin of Saul as having no patience and allowing ‘the disturbing situation to be the most important factor in his decision’. Such an interpretation, which takes the emphasis from the cultic nature of Saul’s impatient action, also resonates with the demand of the holy war that fear be excluded since the nation depends utterly on the victorious presence of Yahweh (cf. Deut. 20.3-4, 8). Similarly, it is possible that the condemnation of Uzziah’s action in 2 Chron. 26.16-20 may not be a statement against the priestly nature of kingship so much as the delineation of a priestly action not permitted to kings in the later Priestly legal tradition (Exod. 30.7-10; Num. 16.40); cf. Rudolph, Chronikbücher, p. 286; J. Becker, 2 Chronik, p. 87. The Chronicler might even here be addressing a specific cultic problem of the illicit offering of incense, given the large number of incense altars excavated from Persian period Palestine (including over two hundred at Lachish alone); cf. A. Rolla, ‘La Palestina Postesilica alla luce dell’archeologia’ pp. 117-18. (Ibid., 58-59, n. 5)
Such is a further refutation of the claim that Psa 110:4 and other priestly texts were about Jesus only and never applied to the Davidic Kings, contra some Evangelicals who forward this eisegetical claim (a claim they forward to defend the thesis that only Jesus is a priest “after the order of Melchizedek” [cf. Biblical Evidence of Multiple People Having Held the Melchizedek Priesthood
Response to The Narrows Church on the Melchizedek Priesthood)
For more on the priesthood, see the listing of articles here, as well as my book, "After the Order of the Son of God”: The Biblical and Historical Evidence for Latter-day Saint Theology of the Priesthood.
