It occasionally happens that as the Masa bakes in the oven, parts of the Masa fold and do not remain flat. This phenomenon is called in Halacha "Masa Kefula" ("folded Masa"). Some people make a point not to eat the folded parts of the Masa, given the concern that the dough in those areas did not fully bake, and thus became Hametz. Those who follow this practice break off the folded pieces of Masa from the rest of the Masa.
Rav Yosef Shalom Elyashiv (contemporary) rules that breaking off a folded piece of Masa is forbidden on Shabbat. The Shabbat prohibition of "Borer" ("selecting") forbids separating "Pesolet" (undesirable matter) from "Ochel" (desirable matter). Breaking off a folded part of the Masa constitutes separating "Pesolet" from "Ochel," which transgresses the Shabbat prohibition of "Borer."
The prohibition of "Borer" applies on Yom Tob, as well, though in more lenient fashion. It is permissible to separate "Pesolet" from "Ochel" on Yom Tob (when it does not fall on Shabbat) if this can be done easily, without too much effort or exertion. Therefore, breaking off a piece of folded Masa, which entails one quick and simple motion, is permissible on Yom Tob. On Shabbat, however, "Borer" is forbidden regardless of the amount of exertion entailed, and therefore on Shabbat it is forbidden to break off a piece of folded Masa.
This would apply as well to breaking off burnt pieces of Masa. If a person does not want to eat part of the piece of Masa which was burnt, he may not break off that part of the Masa on Shabbat. Once again, this would be permissible on Yom Tob, since little effort is entailed, but on Shabbat one may not break off a piece of undesirable Masa.
Rav Elyashiv notes that it is permissible even on Shabbat to remove broken pieces of Masa from the box as one searches for whole pieces. For the Shabbat and Yom Tob meals, one is required to use only whole Masot, and often the box of Masa contains both broken and whole pieces of Masa. Rav Elyashiv writes that one may separate the broken pieces from the whole pieces, as this is not considered removing "Pesolet" from "Ochel." All the Masa is edible, and one removes the broken pieces not because they are undesirable, but rather because he is searching for a whole Masa as required by Halacha. Therefore, he may separate the broken pieces from the whole pieces, even on Shabbat, and this does not violate the prohibition of "Borer."
Summary: It is forbidden on Shabbat to break off from a piece of Masa part of the Masa which one deems undesirable, such as parts that have been folded or have been burnt. This is permissible on Yom Tob. It is permissible even on Shabbat to separate broken pieces of Masa from whole pieces of Masa.
