1) "TERUMAH TEME'AH" IS MUKTZAH
QUESTION: The Mishnah states that Terumah Tehorah may be moved on Shabbos. Terumah Teme'ah, in contrast, is Muktzah and may not be moved on Shabbos.
Why is Terumah Teme'ah Muktzah? It has a useful purpose on Shabbos -- it is fit to be fed to dogs. The Gemara in Pesachim (32a) states that one may feed Terumah Teme'ah which is Chametz to a dog.
RASHI explains that on Yom Tov, a person may not give Terumah Teme'ah to a dog, because such an act is considered a fulfillment of the Mitzvah to destroy Terumah Teme'ah, which one may not do on Yom Tov.
Why, though, is one not permitted to destroy Terumah Teme'ah on Yom Tov? The Gemara earlier in Shabbos (24b) says that one may not burn Kodshim Teme'im on Yom Tov. The Mitzvah to burn Kodshim Teme'im does not override the prohibition of kindling a fire on Yom Tov (when the fire is not made for the sake of food preparation). However, that applies only when there is an Isur d'Oraisa that would be transgressed by burning the Kodshim (such as kindling a fire). What Isur d'Oraisa is there in feeding something to a dog on Shabbos? Why should it be forbidden to give a dog Terumah Teme'ah?
(a) RASHI in Beitzah (27b) explains that the fact that it is a Mitzvah to burn Kodshim makes the act of burning a "significant act," which is considered a Melachah on Yom Tov ("Ach'shevei l'Melachah"). Since the Torah gives this act a high level of significance, it is included in the category of "Melachah" and in the prohibition against performing Melachah on Yom Tov.
(b) TOSFOS in Beitzah disagrees with Rashi. The reason why one may not give Terumah Teme'ah to his dog on Yom Tov is because there is a specific Mitzvah to burn Terumah Teme'ah. (Even though the verse does not specify that the Terumah must be burned, it does mention burning with regard to Kodshim, and Terumah is learned from Kodshim.) Therefore, Terumah Teme'ah is Muktzah because it may not even be fed to a dog, but it must be burned. (Only Terumah Teme'ah which is Chametz, and which must be destroyed, may be fed to a dog.)