I know where you're going here, and it's not very theologically sound.
Leviticus 10 is talking about the temple cult, or in this case the tabernacle cult, this was a sin offering .... (look at the previous Chapter), gaining the favor of Yahweh ...
This offering is going into the house of God offering sacrifice for sin, and/or the favor of God.
So God gave this job to the Levites who were to stay ritually pure, and be the only ones through which God recieved sacrifice, and it was ONLY the High Priest who was to do certain rituals.
When Jesus died the temply cult was finished, which is why, for example, early many Jewish Christians didn't take part in the Jewish cult EVEN THOUGH they kept torah, kosher, shabbat, prayers and so on. This was not an issue of just worshiping God the right way, it was an issue of how sacrifice was to be done, it was to be COMPLETELY pure, and only a Select Group, that was the point of Leviticus 10 ... continue Reading.
6 And Moses said to Aaron and to his sons Eleazar and Ithamar, “Do not dishevel your hair, and do not tear your vestments, or you will die and wrath will strike all the congregation; but your kindred, the whole house of Israel, may mourn the burning that the Lord has sent. 7 You shall not go outside the entrance of the tent of meeting, or you will die; for the anointing oil of the Lord is on you.” And they did as Moses had ordered.
8 And the Lord spoke to Aaron: 9 Drink no wine or strong drink, neither you nor your sons, when you enter the tent of meeting, that you may not die; it is a statute forever throughout your generations. 10 You are to distinguish between the holy and the common, and between the unclean and the clean; 11 and you are to teach the people of Israel all the statutes that the Lord has spoken to them through Moses.
Now the ONLY way that you can apply this to Christianity is if you believe that Jesus sin did not FULLY atone for all sin for all time, period.
Which brings us to the Catholic position .... the Priest is the "alter Christos" and the communion is actual atonement, at least in classical Catholicism, I'm not sure about post Vatican 2, I'm pretty confused about post Vatican 2 theology, but classically, the catholic communion is a sacrifice for sin, it's Christs body and blood, but it really is a continual atonement, rather than once for all time. Now if you believe that, then sure Leviticus 10 applies, but if you don't, and you believe in Pauls' theology, then the atonement is done, there is no more sacrifice, and thus no need for the priesthood or ritual purity or any of the stuff that goes along With sacrifice.