“…He shall then take off his garments and put on other garments, and he shall remove the ashes to outside the camp…” – Vayikra 6:4
וּפָשַׁט אֶת בְּגָדָיו וְלָבַשׁ בְּגָדִים אֲחֵרִים וְהוֹצִיא אֶת הַדֶּשֶׁן אֶל מִחוּץ לַמַּחֲנֶה – ויקרא ו, ד
The task of clearing the ashes from the Altar was not an actual part of the service in the Mishkan and Beis Hamikdash; its purpose was merely to allow more space for the arrangement of wood on the Altar top. This is why the Torah obligates the kohanim to change into garments of lesser value when doing this job, so as not to soil their regular Kohen attire. This was not the only “messy” job in the Temple, but as Rashi explains, “the clothes worn by a servant while cooking a pot of food for his master, he should not wear when he mixes a glass of wine for his master.” Just as cooking takes place “behind the scenes,” in advance of actually serving the master his food and drink, so was the clearing of ash only a preliminary service and not on equal status with the other services in the Temple. It would therefore not be respectful to wear garments soiled by this task, when you are actually serving “before the master.”
Still, this advance work required only a change of clothing, not a change of servant. The very same kohanim who performed the actual service would clear the ash as well. This demonstrates to us that the true kohen—and all of the Jewish people are called “a kingdom of kohanim” (Shemos 19:6)—devotes himself to the preparation for a mitzvah with the same dedication as to the mitzvah itself. The servant of G-d’s primary concern is for G-d’s desire to be fulfilled; he therefore makes no distinction between the roles that ultimately complete the mitzvah and the roles that merely facilitate its fulfillment. He readily immerses himself in the preparations for a mitzvah with equal devotion as to the mitzvah itself.
—Likkutei Sichos vol. 37, pp. 4-5