Q:

What is the word for unconsciously blocking out background noises?

What is the word for the phenomenon where your brain blocks out background sounds, because they are constant?
For example when you don't hear the air conditioning whirring away until it stops.

A:

The word used in psychology for becoming used to a stimulus is habituation

Habituation is a decrease in response to a stimulus after repeated presentations. For example, a novel sound in your environment, such as a new ring tone, may initially draw your attention or even become distracting. After you become accustomed to this sound, you pay less attention to the noise and your response to the sound will diminish. This diminished response is habituation.

It is not limited to sounds (it covers all the senses) but it is involuntary.
Another possibility is desensitise

to make somebody/something less aware of something, especially a problem or something bad, by making them become used to it

Although it does often come with negative connotations, you could use it in a neutral context and be understood.

A:

It's an idiom, "tuning out". He is "tuning out" the noise so he can concentrate on his studies.
For example, when I talk he "tunes me out." Though I am talking to him, he never hears a word I say.

A:

Maybe selective hearing? 
When someone is engrossed in their smartphone and not noticing anything, then they may have inattentional blindness. Is this getting closer?

