To pay no heed to; to not pay attention to or take notice of something.
彼は私の忠告を気にも留めないで仕事を続けた。
He continued working without paying any attention to my advice.
周囲の雑音を気にも留めないで本を読んでいる。
Reading a book without paying any attention to the surrounding noise.
その小さなミスは上司も気にも留めないだろう。
The boss probably won't even notice that small mistake.