I probably didn't notice other people's emotions a lot when I was younger, to the effect that now that I'm older, and do notice them, I frequently don't have any idea what to do with that understanding.
Mostly what I do with all this is just accept it. My feelings are not their problem. Their feelings are not my problem. If I choose otherwise, that's my choice, and so are the consequences. Mostly, I try to take feelings like I take the weather.
That doesn't come easily, though. I feel like I'm learning things that most people learn at 15. Right now I'm working my way through a workbook on DBT, which teaches people emotional regulation skills. It seems crazy to have to sit down and learn things that come so naturally for other people. But then, I'm not other people, so I try to keep pressing ahead.
You should check out the practice in Tsultrim Allione's book, "Demon Feeding: Ancient Wisdom for Resolving Inner Conflict" some time.
Another practice I did for a while to really start examining intense, often unpleasant emotions within myself and others is this: http://www.amazon.com/Feeding-Your-Demons-Resolving-Conflict...
At the end of the day, someone's suffering is their own. By "suffering" I am really talking about dukkha, the Sanskrit word better translated as existential anguish. This is something every person has. It is only when one can clearly see it that they can allow space for someone to work through it. By "witnessing", I am talking about something like practicing vipassana while in the presence of someone suffering. And believe me, that is not as easy as it sounds, neither for the practitioner, or for the person in deep suffering.
Navigating the emotional world is very important and should be taught, not dismissed. As someone who only picked up these skills later in life, its incredible how much life makes sense now. People's emotional responses more or less rule the world. Logic and rationality take a backseat to all of that and being able to gauge other people's emotions and knowing how to respond to them is the difference between failure and success in so many parts of life. Especially if you aspire to any kind of leadership or founder role.
What I was describing is the opposite of dismissing it, and if anything, much healthier. You do realize I was talking about Vipassana and directly experiencing the suffering of others, right?
The method I described is actually quite deep even if on the surface it appears to be simple. It starts with this: http://www.imsb.org/teachings/vipassana-practice/ and might wend it's way through something like: http://www.amazon.com/Feeding-Your-Demons-Resolving-Conflict...