What is the ruling of the Shari'ah on a family remaining attributed to a lineage that they know is not their original lineage because it was changed in the past, and are they sinful for not correcting it due to their current inability to do so?

It is not permissible for a person to change their original lineage and claim descent from other than their father, as this is one of the major sins and leads to grave corruptions. Whoever claims descent from other than his father, knowing full well, then Paradise is forbidden to him. It is obligatory to correct the lineage in official documents, due to the corruptions that result from changing lineage, such as disinheriting an heir and inheriting someone who is not an heir, prohibiting what is permissible in terms of marital relations and permitting what is forbidden, slandering lineage, and disobeying one's ancestors.