What is the ruling on discarding a trash can that contains the name of Allah for someone who dislikes that action but is compelled to do so, and is it sufficient for one who sees the name of Allah in the trash can to merely denounce it in their heart?

It is not permissible to deliberately place anything inscribed with the name of Allah the Almighty in a wastebasket or throw it in the trash, as this constitutes a form of disrespect. The one who does so is not absolved of sin, even if they dislike it in their heart. A Muslim is not obligated to search through trash for such papers, but if they see something sacred, they are required to remove the disrespect from it by hand, either by burning or burying it. Disapproval by heart is not sufficient when it is possible to change it by hand without hardship.