Is the questioner sinful for severing ties with his sister, whose husband swore to divorce her three times and refused to divorce her before the judge, and she refuses to file for divorce for the sake of raising her children, knowing that there are no dealings between them, and is the sister sinful for not filing for divorce?

If the husband has pronounced the triple divorce, then the divorce has become irrevocable simply by his utterance. The judge's role is merely to document and ratify it. The request to go to the judge might be to document the divorce and protect the wife's rights, or for her to obtain the divorce certificate, not for the purpose of enacting the divorce, because it has already occurred. Documenting the divorce protects the woman and does not harm her. It does not prevent raising the children, and the husband is obligated to provide for them. It is not permissible for the brother to boycott his sister because she did not go to document the divorce.