Why some people will never ask for forgiveness?

Because they do not want themselves, they do not have self-esteem, they do not self-respect, therefore, they can not value someone else.

How do we handle friends or people who will never say they feel it?

They are not friends ... they are unhealthy people for your life and mental health.

What is the importance of saying "I'm sorry"?

Just saying "I'm sorry" shows education, culture, common sense, humanity; qualities of great importance nowadays.