Is it obligatory to insist on advising someone who does not respond to it, or is abandoning it after conveying it considered a discharge of one's duty?

Counseling is an obligation for a Muslim towards his brother, and it should be done gently, not in public. As for repeating it, it varies depending on the individuals and circumstances; some benefit from constant reminding, while others are annoyed by it, and it might increase their defiance. Therefore, persistent urging should be avoided to prevent harm, as reminding is done "if the reminder benefits," otherwise not.