Zevachim Chapter 9, Mishnah 5. The Mishnah begins: "Ve'eilu im alu yeredu" - the following items, if they go up onto the Altar, are taken down from it. The intention is not regarding items that are inherently disqualified, but rather items that have no connection to the Altar and no relevance to it whatsoever. Therefore, they should not be brought up there at all, and if they were brought up - they are taken down immediately.
The items that have no connection to the Altar:
"Besar kodshei kodashim" - the meat of a chatas and asham, which is designated to be eaten by the kohanim and not for burning. If it was brought up - it is taken down immediately.
"U'vesar kodashim kalim" - the law is the same: this meat is designated to be eaten by the one bringing the offering, whether shelamim, pesach, or the like, and it has no aspect of burning, and therefore if it was brought up - it is taken down.
"U'motar ha'omer" - the Omer offering brought on the second day of Pesach. As with all menachos, the vessel is filled with an isaron measure, and only the kometz (the handful) is burned, while the remainder is eaten by the kohanim. Therefore, the leftover, what remains after the kometz is taken, is not placed on the Altar, and if it was placed there - it is taken down.
"Shtei halechem" - the two loaves of bread waved with the communal shelamim on Shavuos. It is actual chametz bread, and its entire purpose is to be distributed to the kohanim and eaten by them, and it has no place on the Altar; and if it was brought - it is taken down.
"Velechem hapanim" - the twelve loaves placed on the Table in the Sanctuary throughout the entire week, and eaten by the ministering kohanim. They are not designated for the Altar, and if they were placed upon it - they are removed immediately.
"U'she'ar menachos" - the remainders of the various menachos. As stated regarding the Omer, so too with the other types of menachos, the kometz is taken and burned, and the rest is eaten by the kohanim. If the remainders were brought up onto the Altar - they must be taken down.
"Ve'haktoret" - the incense is burned twice every day, in the morning and afternoon, in the Sanctuary upon the inner Altar, the inner Golden Altar, and not upon the outer Altar. If it was placed upon the outer Altar - it has no connection there, and it must be removed from it.
A new case - items that are not meat:
From here onward, the Mishnah lists additional items that share this law, but one should be precise with its wording: it begins "the wool" without the conjunctive letter "vav," since this refers to a new case. Here it is discussing an olah offering, which is designated to be entirely burned on the Altar after flaying and cutting into pieces. However, the parts that are not the flesh of the animal - the head and the like - do not belong on the Altar in their own right, but rather they come as secondary to the rest of the animal. Therefore, if for some reason those parts separated from the flesh or from the other sacrificial parts (the fats and the like), they are not brought up on their own. This principle of nullification and being secondary is derived from the laws of sukkah.
"Tzemer sheberashei kevasim" - the wool on the head of a sheep or ram. The wool does not belong on the Altar in its own right, but rather the entire head is brought up, and when flaying the animal the head is not flayed (and this too is derived from a verse), and therefore the wool goes up with it. But if the wool separated from the head, it is not brought up onto the Altar on its own.
"Veha'se'ar shebizkan teyashim" - the hair of the goat's beard. It too is secondary to the head, and if it separated - it is not brought up separately.
"Va'atzamos" - the bones go up together with the rest of the meat, but a bone that separated does not go up on its own.
"Vehagidim" - all the long and hard parts inside the animal, including nerves, veins, sinews, and ligaments. A piece that separated and is no longer attached to the meat - its place is not up there, and if it went up it must be taken down.
"U'karnayim" and "u'telapayim" - the horns of the animal and its hooves. These are not meat and do not go up on the Altar in their own right; when the entire leg is brought up, the hoof is attached and goes up with it, but a hoof that is not attached - it is not brought up.
The rule and its reason from the verses:
The Mishnah concludes and establishes the rule: "Bizman shehen mechubarin - ya'alu", meaning as long as they are attached to the leg or to the head, they go up with them, "shene'emar: Vehiktir hakohen et hakol hamizbecha" - the kohen burns and combusts everything on the Altar. But "pirshu - lo ya'alu": once the horn, the hoof, or the bone separated from what is designated to go up on the Altar, they are not brought up, "shene'emar: Ve'asisa olosecha habasar vehadam" - what goes up on the Altar is the meat and the blood, and not horns, hooves, sinews, and bones, and the like.
In summary: In this Mishnah we learned the list of items that if they went up on the Altar - they go down, not because of a disqualification in them but because they have no connection to the Altar: the meat of kodshei kodashim and kodashim kalim designated for eating, the remainder of the Omer and remainders of the menachos, the Two Loaves and the Lechem Hapanim which are eaten by the kohanim, and the incense whose place is on the inner Golden Altar. We also learned the additional case regarding the olah: the parts that are not meat - wool, hair, bones, sinews, horns, and hooves - go up when they are attached, by virtue of "And the kohen shall burn everything on the Altar," and if they separated they are not brought up, as it says "And you shall offer your olah offerings, the meat and the blood."