Meilah, Chapter 3, Mishnah 5. This chapter keeps returning to one question: which items are truly "kodshei Hashem," property of Hekdesh in the full sense, so that deriving benefit from them is an actual violation of me'ilah, and which items carry only a rabbinic prohibition of benefit. Our mishnah applies that question to byproducts: the milk that a consecrated animal produces and the eggs that a consecrated bird lays.
The mishnah begins with two byproducts: "chalav hamukdashin uveitzei torin," what a consecrated animal gives when it is milked and what consecrated doves produce when they lay. Regarding these the ruling is "lo nehenin velo mo'alin": benefit is forbidden, yet a person who did take benefit has not committed me'ilah. This paired formulation is the mishnah's usual shorthand for a prohibition rooted in rabbinic law alone. The items may not be used, but the Torah's law of me'ilah, with everything that follows from it, does not reach them.
Only Regarding Kodshei Mizbeach
The mishnah at once narrows the scope of what it just said: "bameh devarim amurim bekodshei mizbeach." The ruling was stated only where the consecration was made for the sake of an offering upon the mizbeach. In such a case the animal bears kedushat korban. Milk and eggs, however, are never placed upon the mizbeach, and the sanctity of an offering cannot reach anything unfit to be offered. Since they never enter the category of kodshei Hashem, there is no me'ilah in them.
Kodshei Bedek Habayit
Now comes the other side: "aval bekodshei bedek habayit," where the dedication was made toward the repair fund of the Beit Hamikdash, the halachah reverses. "Hikdish tarnegolet, mo'alin bah uveveitzatah." One who consecrated a hen for bedek habayit can transgress me'ilah through the bird and equally through the eggs she lays. "Chamor, mo'alin bah uvechalavah." Consecrate a donkey, and me'ilah is possible with the animal and with the milk she gives alike.
What accounts for the reversal? Nobody imagines a hen or a donkey being set into the stonework of the Sanctuary. The consecration here reaches their worth, not their bodies: the animal is destined for sale, and whatever it fetches is turned over for the maintenance of the Mikdash. Given that this is what the dedication means, eggs and milk stand on precisely the footing of the animal that produced them. They too are saleable, and whatever they bring in belongs no less to bedek habayit.
The Principle
One idea underlies both halves of the mishnah. Where an animal is designated for the altar, what rests upon it is kedushat korban, and that species of sanctity can only take hold of something capable of being offered. Since milk and eggs will never ascend the mizbeach, they never enter the status of kodshei Hashem, and me'ilah with them is impossible. Where an animal is designated for bedek habayit, what was consecrated is its monetary worth, and produce is simply more of that worth. Milk and eggs are funds due to Hekdesh; they are consequently kodshei Hashem, and whoever derives benefit from them has transgressed me'ilah.