(6) The burnt offering (olah) shall be flayed and cut up into sections. (7) The sons of Aaron the priest shall put fire on the altar and lay out wood upon the fire; (8) and Aaron’s sons, the priests, shall lay out the sections, with the head and the suet, on the wood that is on the fire upon the altar. (9) Its entrails and legs shall be washed with water, and the priest shall turn the whole into smoke on the altar as a burnt offering, an offering by fire of pleasing odor to Adonai.
(1) When a person presents an offering of meal to Adonai, his offering shall be of choice flour; he shall pour oil upon it, lay frankincense on it, (2) and present it to Aaron’s sons, the priests. The priest shall scoop out of it a handful of its choice flour and oil, as well as all of its frankincense; and this token portion he shall turn into smoke on the altar, as an offering by fire, of pleasing odor to Adonai. (3) And the remainder of the meal offering shall be for Aaron and his sons, a most holy portion from Adonai’s offerings by fire.
(11) This is the ritual of the sacrifice of well-being that one may offer to Adonai: (12) If he offers it for thanksgiving, he shall offer together with the sacrifice of thanksgiving unleavened cakes with oil mixed in, unleavened wafers spread with oil, and cakes of choice flour with oil mixed in, well soaked. (13) This offering, with cakes of leavened bread added, he shall offer along with his thanksgiving sacrifice of well-being. (14) Out of this he shall offer one of each kind as a gift to Adonai; it shall go to the priest who dashes the blood of the offering of well-being. (15) And the flesh of his thanksgiving sacrifice of well-being shall be eaten on the day that it is offered; none of it shall be set aside until morning.
Why does the Torah find the prospect of leftovers disturbing?
R. Isaac Abarvanel (1437-1508) offers one explanation. The Torah's purpose, Abarvanel argues, is to publicize the miracle the person is celebrating. The fact that a great deal of food must be consumed in a short time leads the thankful person to invite "relatives, friends, and acquaintances to share in his meal and his joy." He will thus have the opportunity to tell them of the wonders and miracles that were done for him, and God's name will be publicly glorified.
I'd like to propose another (perhaps complementary) way of thinking about why the thankful person needs to invite others to share in his or her meal: The nature of gratitude is such that it is inherently outward looking.
Rabbi Shai Held
רמב"ם הלכות יום טוב ו:יח
וכשהוא אוכל ושותה חייב להאכיל (דברים ט"ז) לגר ליתום ולאלמנה עם שאר העניים האמללים. אבל מי שנועל דלתות חצרו ואוכל ושותה הוא ובניו ואשתו ואינו מאכיל ומשקה לעניים ולמרי נפש אין זו שמחת מצוה אלא שמחת כריסו, ועל אלו נאמר (הושע ט') זבחיהם כלחם אונים להם כל אוכליו יטמאו כי לחמם לנפשם, ושמחה כזו קלון היא להם שנאמר (מלאכי ב') וזריתי פרש על פניכם פרש חגיכם.
Rambam, Mishneh Torah, Laws of Yom Tov 6:18
When a person eats and drinks [as part of celebrating a holiday], they are obligated to feed "the stranger, the orphan, and the widow" (Deuteronomy 16:11). But someone who locks the doors of their house, eating and drinking with their children and spouse [alone], and doesn't provide food or drink to the poor and depressed, is not participating in the joy of [God's] commandments (simchat mitzvah) but rather the joy of the gut (simchat kereiso), and about them it says, "their sacrifices are like bread for the dead; all who eat of them will become impure, for their bread is for themselves" (Hosea 9:4) Joy like this is disgrace for them, as it says, "I will strew dung upon your faces, the dung of your festal sacrifices"(Malachi 2:3). [translation by Mechon Hadar]
Rabbi Pinchas, Rabbi Levi and Rabbi Yochanan [said] in the name of Rabbi Menachem from Gallia: In the time to come, all sacrifices will be annulled - but the sacrifice of thanksgiving will not be annulled. All prayers will be annulled, but the prayer of gratitude will not be annulled. This accords with what is written [in Jeremiah 33:11]: "The voice of joy and the voice of gladness, the voice of the groom and the voice of the bride, the voice of those who say 'Give thanks to the LORD of hosts' etc." - this is the prayer of gratitude. "Those who bring [the sacrifice of] thanksgiving to the House of Adonai": this is the sacrifice of thanksgiving. Thus David said: "I owe You vows and will offer you thanksgivings" [Psalms 56:13] - not thanksgiving but thanksgivings [indicating both] the thanksgiving prayer and the prayer of gratitude.
(ג) כִּי שְׁאָר כָּל הַדְּבָרִים יִתְבַּטְּלוּ לֶעָתִיד כֻּלָּם, בִּבְחִינַת: כָּל הַקָּרְבָּנוֹת בְּטֵלִין, חוּץ מִקָּרְבַּן תּוֹדָה (מ"ר צו פ"ט, אמור פ' כ"ז ע"ש), שֶׁלֹּא יִשָּׁאֵר לֶעָתִיד, רַק בְּחִינַת תּוֹדָה וְהוֹדָאָה, לְהוֹדוֹת וּלְהַלֵּל וְלָדַעַת אוֹתוֹ יִתְבָּרַךְ, כְּמוֹ שֶׁכָּתוּב (ישעיה יא): כִּי מָלְאָה הָאָרֶץ דֵּעָה אֶת ה' כַּמַּיִם לַיָּם מְכַסִּים, שֶׁזֶּה כָּל שַׁעֲשׁוּעַ עוֹלָם הַבָּא:
(3) In time to come everything else will be of no consequence whatsoever, as in “All the sacrifices will be abolished other than the thanksgiving-offering” (Vayikra Rabbah 9:7). Of all the hitKaRVut (closeness), the only one that will remain then is the thanksgiving- KoRBan (offering). This is the closeness achieved through thanksgiving and expressing gratitude. Giving thanks and praise and knowing and acknowledging God, as it is written (Isaiah 11:9), “for the earth will be filled with daat (unitive knowledge and awareness) of God, as water covers the seabed”—this is the entire delight of the World to Come.
"...the laws around the consumption of the thanksgiving offering are intended to express and inculcate a core religious value: When one has been the beneficiary of God's kindness, one is expected to bestow kindness oneself. This is both a normative claim–gratitude should be inclusive and outward-looking, and a descriptive, psychological one–true gratitude by its nature is inclusive and outward looking. Otherwise, it is merely pleasure or gladness, but not gratitude."
Rabbi Shai Held