There is a popular Chassidic custom to eat food from which a Rebbe has partaken. This practice is referred to as "shirayim," meaning “leftovers.” The shirayim ritual takes on a number of different forms. In some Chassidic circles, the Chassidim line up to receive wine upon which the Rebbe has recited Kiddush or Havdala. In other circles, the Rebbe has scheduled hours when he receives visitors and distributes fruits or nuts. Yet other Rebbes arrange elaborate gatherings on Shabbat and holidays, known as "tisches," where Chassidim gather around a huge table as the Rebbe eats his meal. In such settings, the Rebbe will often take a small portion of food from a large serving platter, after which the platter is passed around to all the Chassidim in order for everyone to be able to partake of the Rebbe's "shirayim."
The source and significance of this practice is said to originate in a number of sources. Some suggest that it derives from the biblical blessing of plenty, "Blessed shall be your basket and your kneading bowl (mishartecha)" in which the word mishartecha -- with some poetic license -- can be translated as "your leftovers," as if to say: “leftovers [from a tzaddik] bring blessing.” Others suggest that the shirayim custom originates from the episode where Yitro (Moshe's father-in-law) offered sacrifices to God, at which time Aharon and the elders came to eat with him. Commenting on this verse, Rashi quotes the Talmud which teaches that, "one who derives pleasure from a meal which includes Torah scholars is as if he derives pleasure from the Divine Presence."
Another approach has it that the shirayim concept is an extension of the mitzva of "peah" in which one must leave a corner of one's field for the poor. In this case, the mitzva of peah is “extended” to include leaving over some food at a meal for others to enjoy. Indeed, the Talmud teaches that "one who does not leave over bread at his table will never see a sign of blessing." Hence, shirayim, whether consisting of bread, fruits, nuts, or other foods, is the "leftover bread” which is said to bring blessing.
Rav Shlomo Aviner relates that Rav Avraham Shapira believed that the shirayim custom derives from the Talmud, which teaches that it is forbidden to eat from a loaf of bread that a mouse has eaten from. This is because a spirit of impurity spreads throughout such bread. It is reasonable to suggest, therefore, that when a righteous person eats from bread (or any other food for that matter), purity and holiness are spread throughout it. Rav Moshe Halbershtam was said to have acknowledged this as possibly being the primary source for the shirayim custom.
The Yerushalmi relates that Rav Yochanan would go to the synagogue each morning to collect the leftover crumbs that remained from a seudat mitzva that may have been held the night before. As he ate them he would say, "May my portion be like those who ate here yesterday." It is taught that just like Rav Yochanan would eat the leftovers of a seudat mitzva with great enthusiasm, so too the leftovers of a tzaddik, and his seudat mitzva, should be eaten with great enthusiasm, as well. There are also Talmudic sources that teach that when food is distributed by the head of the household, it brings blessing to the family. In this context, of course, the Rebbe is the head of the “household” while the Chassidim are “the family.”
Rebbe Yitzchak Isaac of Zhiditchov said in the name of the Apter Rav and Rebbe Elimelech of Lizensk that a tzaddik leaves his mark on the food he eats. Therefore, his shirayim contain his tzura, his form or “presence,” and are holy. The Komarna Rebbe commented on the verse, "And he took portions to them from his own table (masos me'et panav)…" relating it to an episode in which Rebbe Zusha of Anipoli noticed that his brother Rebbe Elimelech’s tzura was in his home. When Rebbe Zusha saw his servant carrying shirayim from his brother's meal, he realized that it was through the shirayim that his brother’s tzura entered his home. Thus, he creatively interpreted this verse to "prove" that food can contain the face (panav) of a tzaddik. Eating the leftovers of tzaddikim is said to be comparable to eating from the kodshim, sacrifices that had been offered in the Beit Hamikdash. Such food is said to taste better due to the holiness that rests upon it.
Finally, there is a concept within Chassidic thought known as "birur hanitzotzot" which teaches that when a tzaddik touches something he is able to unify all the Divine forces within it, thereby elevating the object to a higher spiritual level and purifying it. Related to this is the belief that food actually contains the sparks of transmigrating souls. When one consumes such food it allows these “soul sparks” to be released and move freely. The souls, in turn, bless the one who ate from the food. According to this approach, this "soul release" is more assured of being accomplished from food that has been elevated by a tzaddik’s blessing.
 Devarim 28:5.
 Sefer Chassidim 888.
 Letters from the Rebbe, Vol. 3 p. 193.
 Shemot 18:12.
 Berachot 64a.
 Vayikra 19:9-10.
 Sanhedrin 92a.
 Chagiga 13b.
 Rosh Devarcha p. 108.
 Moed Katan 2:3. See also She’arim Metzuyanim Behalacha 42:2 and Derech Sichah vol. 1 p. 220. Interestingly, the commentators teach that the seudat mitzva in question was the Seudat Ibbur Hachodesh, the meal that would be held in honor of the Beit Din having added an extra month to the calendar.
 Bereishit 43:34.
 Eser Tzatzachot 73, cited in Mipeninei Noam Elimelech p.258.
 Tiferet Shlomo, Re'eh.