Is it permissible to lie under oath to conceal oneself from a past sin, and does this fall under the concealment mentioned in the Sacred Hadith: "All of my Ummah will be forgiven on the Day of Resurrection except those who make their sins public"?

A Muslim, if they commit a sin, is required to conceal it, unless there is a benefit in disclosing it, such as seeking a fatwa (religious ruling) or advice. This is based on the Prophet Muhammad's (peace be upon him) saying: "All of my Ummah (nation) are forgiven except those who publicize their sins..." No one has the right to demand an oath from another person that they have not committed sins, except for a judge under specific conditions. A man swearing falsely that he did not commit adultery or drink alcohol is a "Ghamus oath" (a false oath that plunges the swearer into sin), and its expiation is sincere repentance.