I want to describe something like below,
I can't ignore the emotion on people's face, I mean, those people who are important to me.
But it seems like I'm a cruel person, that's not true, I found a stranger in a sad mood, I would like to comfort him or her, to make she or he feel better, but here I want to stress that people who are important to me, I value their emotion very much.
So guys, how to say my meaning in a concise and accurate way?
If you say
I can't ignore the emotions on the faces of people I care about.
it is more succinct than your original, but a listener may still infer that there are people outside that group. That's how restrictive clauses work: they divide the world into the things that match the description and, by implication, those that do not.
If you say
I can't ignore the emotions on people's faces.
It's a general statement without restriction in regard to people, so it is silent with respect to the idea that some people are more important to you than others.
You cannot have it both ways, unless you say something like :
I can't ignore the emotions on people's faces, especially the people I care about most.
This is only partially a question of language.
First, as a purely linguistic matter, you should look up the difference in meaning between "empathy" and "sympathy." It is not clear which you intend or whether you intend both.
Second, you seem to be seeking a purely linguistic solution to what are moral questions, namely whether we morally owe both strangers and friends an equal degree of empathy and sympathy, and, if so, can humans meet that purported obligation.
You can say
I have great empathy with and sympathy toward others.
That sentence makes no admission of moral fault (if fault there be) at the expense of hiding the truth about your personality.
You can say
I have great empathy with and sympathy towards those dear to me.
That sentence is forthright about your personality at the expense of admitting possible moral fault
If you personally care more for those dear to you, language lets you refuse to acknowledge that aspect of your personality and thus evade certain moral questions. Language also lets you acknowledge that aspect of your personality but thus raises those moral questions.
Linguistically, choose between "empathy" and "sympathy," but don't expect linguistic magic to resolve moral questions.
