The Taz and Magen Avraham at the end of OC 167 bring from the Abudraham that you should not give animals (or goyim or birds, says the Mishna Berura,) the prusa of hamotzi, the bread upon which you had made hamotzi. So what do you do with it? Make a bread kugel, or put it into the cholent next week, I guess. The Taz there says that the Medrash Rashi quotes on our possuk is a good source for the Abudraham’s idea; if the concept of chivuv mitzvo goes to the extent that they carried the leftovers in their hands and did not allow their animals to carry them, then kal vachomer one should not feed the leftovers to animals.
We find a similar idea here: the Gemora in Brachos 39b and Shabbos 117b says that Rav Ami and Rav Assi, when they would have the Eiruv Chatzeiros bread available, would use that for Hamotzi. “ רב אמי ורב אסי כי הוה מתרמי להו ריפתא דערובא מברכין עליה המוציא לחם מן הארץ אמרי הואיל ואתעביד ביה מצוה חדא נעביד ביה מצוה אחריתי ” once it has been used for one mitzvah, let us do another mitzva with it. In fact, the Shibalei Haleket at the end of 246 says you should davka use a shaleim, an unbroken loaf of bread, for eiruv, so you will be able to use it for hamotzi. We see that an item that is left over from being used for a mitzva, even a relatively very minor mitzva such as Eiruv Chatzeiros, should be shown respect and used in the most honorable manner possible.
Of course there is a difference between leftover matza from the Korban Pesach of Yetzias Mitzrayim and leftover bread from hamotzi. Simply saying Hamotzi on a piece of bread is no different than saying Shehakol on a glass of water; we don't find anyone that says that you have to use the leftover water for a mitzva. But in truth, you don't need a makor for this din. As we see from Rav Ammi and Rav Assi, it's a simple svara-- show respect to an object that was used for a mitzva. You don't have to put your leftover Cholent into Geniza. But when you have the opportunity to use it in doing another mitzvah, you should do so. Hamotzi, apparently, is a more significant bracha than others, in that there is a hava amina in Brachos 35 that it is Mid'oraysa; so bread that you made hamotzi on is considered a cheftza shel mitzva.
Please note that these ideas are primarily minhagim; the Gemara in Megilla 26b allows us to throw away items used for Mitzvos. It is on the authority of the Shi'iltos and the Kol Bo and the Maharil that we are machmir beyond the literal meaning of the Gemara. The more commonly known application of this hanhaga is for items that are specifically for mitzvos, which have no meaning outside of the mitzva, and which have some special preparation for that use, such as making them Lishmah. For example, in OC 21 we learn that tzitzis don't need geniza, but should not be subjected to disrespectful use, and even the garment to which the tzitzis were attached should not be used in a degrading manner. Similarly, this is brought in Rama 664:9 on Hoshanos and in Mishna Berura 297 SK 8 on using the Hadas for Besamim for Havdala, and see Magen Avraham in OC 638 SK 2 regarding Schach. The way the idea is used here is far broader than the those cases. Also see Mishna Berura 673 SK 27 to use the wax that dripped from the candles in Shul for Chanuka, and Ritva Shabbos 117b that if you used a pas for hamotzi once, and it is still shaleim, (I guess you used a different piece for eating) you should use it for hamotzi another time, as brought by the Mishna Berura in 527 SK 11 and 48, that bread that was used for eiruv should be placed with the bread for hamotzi and eaten at Seuda Shelishis.
This issue arises in the discussion regarding inviting non-Jewish friends and colleagues to the Seder. It should be obvious that there are many issues that should be discussed on this matter. Besides the basic incongruity of such guests at a seder, about which the Torah says זאת חוקת הפסח כל בן נכר לא יאכל בו, and the oddity of saying and explaining Shfoch Chamascha, one of the more serious problems is the Halacha discussed in OC 512, and see Mishna Berura there SK 6. There is also the matter of teaching Torah to non-Jews, as in Igros YD 2 :132. Of course, an entirely different perspective comes into play if the guest is in the process of conversion, as discussed in Minchas Elazar 3:8. But directly relevant to this question of non-Jewish guests, and giving Matza and items from any Seudas Mitzva to non-Jews, is the following Kaf Hachaim on Siman 167, starting with SK קמ:
(By the way, many people have the habit of never throwing out bread; they feel that disgracing bread by putting it into the garbage can bring poverty, as indicated in the gemara (Chulin 105b) about the guy who made a picnic and gathered the crumbs. I have a neighbor that throws all of their leftover bread into the alley, the small street behind our houses used for deliveries. It's ironic: to avoid bizayon of bread, they throw it where it will be stepped on and ground into the mud. This is not only stupid but also entertaining. On the other hand, they're still very wealthy, so maybe they're right and I'm wrong.
Another local oddity- a friend, Rav Shimon Kalman Goldstein, told me that he knows a nursing home operator who gives out hand shmura matzos to gentile business acquaintances before Pesach. SK has told him that based on the Kaf Hachaim, this might be a misuse of Matza Shmura, but it didn't seem to have influenced that person. I suggested that he most likely doesn't eat gebrokst, so he probably sprinkled them with water before giving them away and 'deconsecrated' them. Reb Dovid Kronglas once said that every midda can be used for avodas Hashem. How do you use the midda of krumkeit? In being דן את כל האדם לכף זכות. Usually, to be dan people l'kaf zechus, you need to use the midda of saying a krumeh svora.)
A reader wrote in a private correspondence that he once attended a shiur by Rabbi Heinemann on the topic of respect for items used in fulfillment of mitzvos, and another listener asked whether married couples need to show kavod to their mattresses in light of the mitzva of Piriah ve'Rivyah. I assume the audience at that moment thought it a humorous question. However, I would say that the concept applies there no less than it does to bread used at the table on Shabbos of to say hamotzi on. It is, after all, הלחם אשר הוא אוכל, and while some people overeat, or eat with bad manners, or eat because they enjoy eating more than they desire to honor Shabbos with Lechem Mishna, it's still called a cheftza shel mitzva. I would have answered that we are only mechabeid things that are used in fulfillment of mitzvos, while in that case, the item was used in the ma'aseh mitzva, not the fulfillment of the mitzva. Which, of course, raises the question of whether it is a ma'aseh mitzva or a real kiyum hamitzva, and children are just the shiur for petur. This is more something Chaim B. would address; see, e.g., here, third paragraph.
Chaim disagrees in the comments. He says that the bed is no more cheftza shel mitzva than the table at which you're eating the Shabbos meal. To this I respond with Rabbeinu Bachaye in Parshas Teruma, who, in explaining the pasuk in Yechezkel 41:22, 'זה השולחן אשר לפני ה, tells us that the tzadikim of France would be buried in an aron made of the wood of their dining room table. You want to be buried in an object that served you in your kiyum hamitzvos. I remember Rabbi Weinberg telling me that he wanted his shtender, from which he said shiurim, to be buried with him. Unfortunately, when I told this to his family, they didn't believe me. He probably should have told them himself, or left instructions in his tzava'ah. I wonder what I would want buried with me....the computer mouse? Not a good idea. Maybe a daf yomi calendar. Or torah from my children and grandchildren.