"I don't give a damn" means you do not care at all about something or someone's opinion. It is blunt, informal, and can sound rude or dismissive, showing strong indifference.
I don't give a damn for man that can spell a word only one way. Mark Twain
I'm intact, and I don't give a damn. Arthur Rimbaud
I don't give a damn about "The Missouri Waltz” but I can't say it out loud because it's the song of Missouri. It's as bad as "The Star-Spangled Banner” so far as music is concerned. Harry S. Truman
And I don't give a damn about a greenback dollarSpend it fast as I can for a wailin' song and a good guitarThe only things that I understand, oh yeah. Hoyt Axton
When I get my hands on painting materials I don't give a damn about other people's painting... every generation must start again afresh. Maurice de Vlaminck
I reserve the right to tell shaggy dog stories or even common jokes as part of what I'm doing. I don't give a damn if half the audience walks out. David Antin