The Lisker Rebbe zy”a (Sefer Ach Pri Tevuah) quotes Sefer Noam Magadim as explaining the words “you shall not become tamei to a nefesh in your nation” to mean that one should not find fault in any Jew.
The pasuk states (Bamidbor 12:3): “And the man Moshe was extremely humble, from any man on the face of the earth.” This means that whenever Moshe saw any Jew commit a sin, he blamed himself because he knew that he was the “shoresh” of every Jewish soul. Just like the branches of a tree grow from its roots, so too all of Klal Yisroel grew from Moshe.
Similarly, the pasuk is saying to tell the kohanim that they shouldn’t consider any nefesh in Yisroel “tamei b’amav”, meaning that if anyone in am Yisroel seems impure, they shouldn’t blame them. Rather, they should see it as “l’she’eiro hakarov eilav”, to the relative close to him, meaning that they should blame themselves and say that it must be their fault that someone is acting that way.
The pasuk is explained similarly by Sefer Duda’im B’Sadeh (written by a student of Rav Elimelech of Lizhensk zy”a). He writes that in order to properly serve Hashem, one must be exceedingly humble in his own eyes and never look down on others. One must recognize that everyone Is expected to serve Hashem as well as he can according to his level of understanding. It is possible that a simple Jew is doing his best and fulfilling his limited potential and, therefore, is more worthy than a tzadik, who may not be reaching his full potential of greatness. This is how Moshe was a bigger anav than any other man, as he always considered his avodah as being not as great as it could be, while he looked at others as fulfilling their full potential in avodas Hashem.
He thus explains that kohanim are commanded not to consider any Jew to be tamei, as they should see everyone as trying to do the best they can. He also uses this idea to explain Rashi’s words “l’hazhir gedolim al haketanim”, meaning that the gedolim, i.e., the tzadikim, are commanded to consider the “small people” to be tzadikim.