Zevachim chapter 9, Mishnah 3. The Mishnah opens: "Eilu lo hayu pesulim bakodesh" - The disqualifications listed here are not disqualifications that occurred during the sacrificial service, but rather existed beforehand, even before the animal entered under normal circumstances through the gate of the Azarah. Therefore, they are completely disqualified: they have no business being offered, and if they went up - the limbs, the emurim, and the sacrificial parts are taken down from the altar, since they are considered a disgrace to the altar.
And these are the disqualified animals:
"Harovea vehanirba" - An animal that participated in an act of bestiality, where the rovea is the male and the nirba is the female. This abominable act disqualifies it from being offered, and if it was offered - it is taken down.
"Vehamuktzeh vehane'evad" - 'Muktzeh' is an animal that was set aside to be offered as a sacrifice for idol worship, and 'ne'evad' is an animal that was itself worshipped as an idol. These too are completely disqualified.
"Ve'esnan umechir" - 'Esnan' is the payment given to a harlot, and 'mechir' is an animal exchanged for a dog. A harlot cannot bring the goat she received as her payment as a sacrifice, nor can she transfer it to a third party, to sell it to someone else who would offer it. The same applies to mechir: one who exchanges a dog for a goat no longer has the dog in his possession but rather the goat, and nevertheless, he cannot offer it, nor can he sell it to another for a sacrifice, because the animal is permanently disqualified due to the use made of it to purchase a dog.
"Vechilayim" - An offspring born from two different species, such as a goat and a sheep that mated and produced offspring. This is very far from common, but it has happened, and such a creature is forbidden to be offered.
"Utereifah" - An animal that suffered a fatal injury of any kind, and it will not continue to live.
"Veyotzei dofen" - An animal that was not born through natural birth, but rather was extracted from the side of the animal via surgery, similar to a cesarean section.
"Uva'alei mumin" - Animals that sustained a blemish, which also disqualifies them.
All these things have no place going up on the altar, and if they already went up on it - they must be taken down.
The leniency of Rabbi Akiva:
The Mishnah brings the opinion of Rabbi Akiva concerning blemished animals: Rabbi Akiva permits the emurim of a blemished animal to remain there, after they already went up on the altar. The Gemara explains that this refers to minor blemishes - of the type that if they were on a bird, it would be permitted to offer it initially, things that are barely noticeable, such as a cataract in the eye and the like. It does not refer to a three-legged animal, which certainly has no place going up on the altar at all, from the outset.
The testimony of Rabbi Chananya segan hakohanim:
Rabbi Chananya segan hakohanim served in the Beis Hamikdash as second to the kohen gadol, and from his practical familiarity with the order of the service, he testifies: "Docheh hayah abba ba'alei mumin me'al gabei mizbeiach" - He had a tradition from his father that he would indeed remove from the altar parts of blemished animals that went up on it, except that he used the term "docheh", meaning pushing aside in a modest way. That is, he did not make a big deal out of the matter and did not act publicly, in an open way where everyone sees what is happening, but rather secretly and modestly, so as not to attract much attention - and all this out of respect for the Beis Hamikdash. Regardless, even according to him, they remove blemished animals, sacrificial parts of blemished animals, if they ultimately went up on the altar.
The halachah: All these disqualified animals, which existed even before the beginning of the sacrificial service, have no place going up on the altar, and if they were brought up - they are taken down, even if they already went up on the fire. And with a blemished animal of the minor, less severe type, they take it down modestly and discreetly, not publicly and without a fuss, in order to preserve the honor and splendor of the altar of the Beis Hamikdash.