The Torah teaches about the daily sacrifices in the Mishkan. Every day before dawn, we are told, (3) And the kohen shall put on his linen garment, and his linen breeches shall he put upon his flesh, and he shall raise up the ashes of the burnt-offering that the fire consumed on the altar, and lay them at the side of the altar. (4) Then he shall put off his garments and put on other garments and carry forth the ashes outside the camp to a pure place (Vayikra 6).
The ashes were removed even before the first daily sacrifice (tamid shel shachar), and before arranging the wood for the fire (ma'arachah). A kohen, selected by lot, would enter the Sanctuary that was lit only by the still-glowing fire on the altar. First, he immersed himself in a mikveh, and sanctified his hands and feet - as before every divine service (Avodah) - to perform the terumat ha'deshen, offering some of the ashes of the sacrifices brought the day before and which had burned throughout the night.
Although this was a messy job - and thus the kohen wore simple clothes, albeit of the four priestly garments (Yoma 24a) - many kohanim would flock to the Sanctuary to be chosen by lot for this task. Terumat ha'deshen is an Avodah, and it was an honor to be selected. (Clearly, it was not beneath the kohen's dignity to get a bit dirty in the service of Hashem!)
Moreover, the services of the Mishkan could not proceed without terumat ha'deshen: If the ma'arachah had been mistakenly set up beforehand, the wood pile would have to be disassembled so that the terumat ha'deshen could be done first (Yer. Yoma 2:1). Only afterwards would the other kohanim begin the services of the new day.
The mitzvah of clearing the ashes, however, is more than merely cleaning up the altar. Actually, our Sages identify two activities in these verses:
The Jewish "Today" has to take its mission from the hand of its "Yesterday." Actually, the most perplexing part of the terumat ha'deshen is where it is placed - in the earth. The general principle is that we always elevate levels of sanctity rather than descend (ma'alin b'kodesh v'lo moridin; Berachot 28a). One would expect therefore, that we should raise the sanctity of the ashes, in keeping with the verse itself: and he shall raise up the ashes. Instead, we lower it to the earth, to be absorbed into the ground! However, we are commanded, lay them at the side of the altar.
In fact, this placing in the earth is its elevation. Rather than being a step downward, placing the terumat ha'deshen in the earth is actually a step up. Reb Meir of Przemyslany (died, 1773) owned a cow. Every week he would sell the milk and distribute the money to the poor. However, one Wednesday, the poor were particularly in need, so he had the cow slaughtered and distributed the meat. When his wife came home and could not find the cow, she exclaimed "Our cow is lost!" "No, dear wife," answered Reb Meir, "it has gone to heaven!" There is no higher sanctity than when the values of the Sanctuary permeate our daily life. The goal of the sacrifices - indeed, the goal of all acts of holiness - is for the ideals of sanctity to be absorbed into our everyday existence. Rather than withdrawing into the Sanctuary, our task as Jews is to make this world into a sanctuary.