The shells of nuts, such as pistachio nuts, are considered Mukse on Shabbat, since they are not edible for humans or animals. This does not mean, however, that it is forbidden to eat nuts on Shabbat. Since the nut contains both a shell and the fruit inside, it is certainly permissible to take nuts and eat them on Shabbat. Furthermore, after one finishes eating, and the shells are in his hand, he is allowed to carry them to the place where he wishes to discard them. For example, if he is eating in the dining room, and the shell in his hand, he may carry the shell to the kitchen to throw it in the trashcan.
One may not, however, transfer the shell from one hand to the other. Transferring the shell to a different hand would constitute "Tiltul Hadash" (a "new carrying"), and is therefore forbidden on Shabbat. This is the ruling of the Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) and the Kaf Ha’haim (Rav Yaakob Haim Sofer, Baghdad-Israel, 1870-1939). By the same token, it would be forbidden to give the shell to somebody else to throw out. For example, if somebody is leaving the table and going to the kitchen, one may not hand the shell to that person and ask him to take the shell to the garbage for him.
Many people are unaware of the fact that olive pits are also considered Mukse on Shabbat, as they, too, are not edible for human or animals. Of course, it is permissible to eat olives on Shabbat, because at that point it still has fruit, which is not Mukse. After one eats all the fruit of the olive, however, and only the pit remains, the pit becomes Mukse and may not be handled. The question thus becomes what one should do on Shabbat if he eats an olive and the pit is in his mouth. If he removes the pit from his mouth with his hand, this would constitute a "Tiltul Hadash," and he would be in violation of handling Mukse. The Gemara therefore relates that the Sages would follow one of two procedures to avoid the Mukse prohibition when eating olives. The first possibility is to ensure not to eat all the fruit of the olive. If some fruit remains on the pit, then one may remove it from his mouth with his hand, since it has not yet become Mukse. Alternatively, if one ate all the fruit and nothing remains on the pit, he may simply spit the pit into his plate. (Of course, this might not follow the highest standards of etiquette…) Since the pit is Mukse, he may not handle it with his hand, and he therefore has no choice but to spit it directly onto his plate.
Summary: If one eats a nut on Shabbat and has the shell in his hand, the shell is considered Mukse but he may keep it in his hand until he comes to the place where he wants to discard it. He may not, however, transfer it to the other hand or give it to another person. If a person eats an olive on Shabbat, he may not remove the pit from his mouth with his hand, unless some fruit of the olive remains on the pit. If no fruit remains, then he has no choice but the spit out the pit directly onto his plate.
