Bechorot 14 - Consecrating Animals with a Blemish
An offering must be without blemish, and even consecrating a blemished animal is forbidden. Actually, if one consecrates a blemished animal as a sacrifice, no consecration occurs at all, but if he says, "I am consecrating this blemished animal in order that it be sold, and a sacrifice brought with the money," then, although he violated a prohibition, the animal needs to be redeemed.
After redemption it is again liable to the law of firstborn (that's the connection of this ruling to "Bechorot"), it may be shorn and put to work, and its offspring is permitted. Now, if the offspring was born after redemption, obviously it is permitted, and if it was born before the redemption, it should be forbidden, because it is unblemished, fit for a sacrifice! However, its law cannot be more strict than that of its mother's - who is permitted.
If it dies, its carcass may still be redeemed. But usually for redemption the animal needs to stand up, which a dead animal cannot do! However, this is the opinion of Rabbi Shimon, who requires standing only for real sacrifices. Those who disagree with him would indeed disallow redemption and require the animal to be buried.
Art: William Gowe Ferguson - Still-Life with Dead Game