In regards to the halacha that a piece of bread that is at least the size of a kezayis must be covered and thrown out, is it sufficient to wrap the piece in a napkin or does it need to be placed in a separate bag?
The idea of not throwing out bread or other food that is edible is, is because we are not allowed to directly destroy food, and also because it is disrespectful and embarrassing for the food. The food may be left to rot, as this is not directly destroying it. A number of Poskim say that throwing food directly into the garbage, is considered directly destroying it, however if it is wrapped up, it may be thrown into the garbage. Therefore as long as it is wrapped up in a way that it isn’t mixing with the garbage, such as a napkin or bag, that is sufficient.
Shut Even Yisrael 9-62 (11), Vezos Ha’berachah page 18, Eitz hasadeh 35-4,5, also see ftnt. 13 in the name of R’ Eliyashiv zt”l and other poskim, also see ftnt. 16 in the name of R’ C. Kanievsky shlit”a. Note- There are poskim who say that if possible the food should not be thrown out if it is still edible to humans, see Mishne Halachos 15-64, Chut Shani (Shmitta V’yovel) pg. 289..