Ignoring something:
* Turn a deaf ear to: This is the closest idiom to "turn deep ears to something", meaning to deliberately ignore something.
* Not bat an eyelid: To show no surprise or reaction to something.
* Not give a hoot: To not care about something at all.
* Not hear a word: To completely ignore what is being said.
* Let it go in one ear and out the other: To not pay attention to something.
* Not notice a thing: To be completely oblivious to something.
Paying no attention:
* Have your head in the clouds: To be lost in thought and not paying attention to what is happening around you.
* Be oblivious to: To not be aware of something.
* Be distracted: To have your attention drawn away from something.
* Be in a world of your own: To be so preoccupied with your own thoughts that you are not aware of your surroundings.
The best idiom to use will depend on the specific context of the situation.