Although there is a prohibition of possessing chametz on Pesach, you probably have no need to feel guilty about not finding all the chametz. If you are referring to insignificant crumbs, you need not worry, since they are not included in this prohibition.[1] Regarding significant pieces, you still don’t need to worry, as long as you followed the prescribed pre-Pesach procedure of cleaning, culminating in bedikat chametz, and bitul (nullification of) chametz. This is so for the following reasons: 1. One who checked properly (as can be expected of a human) has fulfilled his mitzva and does not violate the prohibition of possessing chametz, even if some chametz remained unnoticed. This is either because it is an oness (beyond control) or because, according to some, one violates the prohibition only with chametz that he knows about.[2] 2. The Rabbis were concerned, even after a person has performed bedikat chametz, that he may discover chametz in his possession and, at that point, will violate the prohibition of owning chametz. Therefore, they instituted bitul chametz, whose text can be found in siddurim and in bedikat chametz kits. Once bitul has been done, one no longer violates the prohibition of owning chametz, even if he finds chametz in his possession on Pesach.
Still, if one finds chametz on Pesach, he may not rely on the bitul but should destroy it at the first opportunity.[3] If one finds chametz on Yom Tov, it is muktzeh and it should be covered with a utensil to avoid the possibility that it will be eaten accidentally.[4]
The prohibition, after Pesach, of benefiting from chametz that was in Jewish possession on Pesach (chametz she’avar alav haPesach) is a rabbinic injunction as a penalty to those who violated the prohibition of possession.[5] To strengthen the injunction, it was applied even to cases where the owner of the chametz is not to be blamed for the mishap or even when he did bitul.[6] Thus, if one finds (unsold) chametz in his house after Pesach, he should throw it out. If this will cause great loss, he should consult his local rabbi.[7]