The skins of the kodshei kodashim, the holiest korbanot (offerings), were given to the Kohanim (priests). Anything from an olah (burnt offering), chatat (sin offering), or asham (guilt offering), and other korbanot, would go to the Kohanim. After the korbanot were slaughtered and the blood was sprinkled on the Mizbeach (altar), they would skin the animal, take the portions that were supposed to be burned on the Mizbeach, and then give the skins to the Kohanim.
If the korbanot became disqualified, the halachic status of the skins depended on the point in time at which the disqualification took place. If the korban became disqualified while the skins were still on it, then the skins became disqualified as well, and they would have to be burned together with the rest of the korban on the Mizbeach. But if the korban became disqualified after the skins were removed, the disqualification doesn't affect the skins whatsoever. Rather, they give the skins to the Kohanim.
Here we'll talk about a situation where the korban became disqualified while the skins were still attached, but they only discovered the disqualification after the skins were removed. The Mishnah says, Amar (said) Rabbi Chanina, who was an officer of the Kohanim: “In all of my time, I never saw a skin taken out on its own to be burned in the place of burning.”
What Amar Rabbi Chanina is basically testifying about is that once the skins were removed from the korban, they were never taken to be burned. So we see from here that after the skin was removed, it's never burned under any circumstances, even if the korban is later discovered to have been disqualified all the way from the start, even while the skin was still on it. So, from his testimony, it seems to be about the sha'at ha-metzi'ah—in a way, when do you find out about it? If you found out about it afterwards, now that the skins are already off, we don't burn them, but we allow the Kohanim to keep them.
Rabbi Akiva applies this to a very similar situation. The skins of a blemished Bechor (firstborn animal) are used by the Kohanim unless the disqualification happened at the time of Shechita (slaughter), or was discovered at the time of Shechita. So, if the Shechita was assumed to be valid at the moment it happened, and then later it was found to be invalid, Rabbi Akiva says that from the words of Rabbi Chanina, from his idea, we could learn—we learned that if we slaughter and skin a blemished Bechor, and then later on it was found to be a Terefa, that the Shechita was not done correctly, the Kohanim could still use the skins.
The Shechita at the time was assumed to be valid when it was done. So, if the Shechita was okay, then the skins are also permitted forever, even though later on we find that the Shechita was discovered to have been invalid. If we think it was okay at that time, and when we skinned it, we assumed that everything was okay, and then later on we find that it's a problem, since at the time we found that it was an issue, the skins were off, we are allowed to use the skins as well.
The Chachamim are going to disagree with Rabbi Chanina's earlier ruling. The Chachamim (sages) say, “The fact that he never saw it is not a proof.” Just because Rabbi Chanina never saw the skins being removed, or taken out on their own to be burned, doesn't prove what he was trying to say: that there's never a situation where the skins are not allowed to be used if they were off the korban at the time they found out that the korban was an issue. The fact that he never saw it doesn't mean that it didn't happen. Maybe he never saw it himself, or maybe the situation never arose, but if it arose, of course it would be a problem.
The halacha (Jewish law) is that the skins of a korban that was disqualified are taken out to the Beit HaSereifah, the spot of burning, even if the disqualification was discovered after the skins were already removed.