Of the 613 mitzvot, some are applicable only to kohanim. These mitzvot are what elevates the kohen closer to g-d and are viewed as an indicator of the koach kohanim have to fulfill these mitzvot.
שלוחי דרחמנא נינהו
Rashi translates kohen as משרת – servant, ok, but whom does the kohen serve? this question is actually debated in Talmud Bavli (Nedarim, Kidushin, Yoma); whether the kohen’s is a servant of g-d or of the jewish people. With the conclusion of that debate being that the kohen is a servant of the g-d of Israel.
The kohen’s service to g-d is unique in the sense that essentially, all of mankind serve g-d, whereas the torah relays that g-d chose the Jewish people as his “treasure” from among the seventy nations of the world. From the twelve tribes of Israel, the tribe of Levi for was chosen for specific services. From the tribe of Levi, g-d elected the descendants of Ahron as personal servants for holy duties, as is found in divrei hayamim: ויבדל אהרן להקדישו קודש קדשים “And Ahron was separated, to sanctify him, holy of holies”.
What the mishna tells us about Ahron is essentially one of the primary duties of the kohen;
Be of the students of Ahron, loving peace and chasing peace, loving the creations and bringing them close to the torah
This mishna teaches that the kohen serves g-d as his forefather Ahron, loving shalom and bringing people close to the ways of the torah. But herein lies a key phrase “bringing them closer to the torah”, this phrase echoes how so the kohen is a servant of g-d, where although on the one hand his heritage breeds a powerful attraction to peace, this attraction isn’t to come at the compromise of torah law but by “bringing the person close to the torah” and not “the torah close to the person” (i.e. by compromising torah law see תורת מנחם חלק יז. תורת מנחם חלק כ).
Sounding silver trumpets
Kohanim are commanded to sound silver trumpets at times of calamity (cha.”v).
The kohen is obligated to view nega’im of various types, as only the kohen may pronounce an afflicted person, garment or home as tammei. However, only these types of tumah require the kohen, other types of tumah such as zav, zavah etc. do not require a kohen.
Of the important duties of the kohen is torah instruction to the nation of Israel. Although the duty for the kohen to instruct torah is not listed amongst the 613 mitzvos, it is a theme quoted in the torah and by chaza”l.
The duties of the kohen are many even in the absence of the bet hamikdash (to be speedily rebuilt). They include;