Hi 김시온 I wonder if the questions are incomplete? If you ask "How do you feel about that?" or "What do you feel about that?", then they both mean more or less the same and are interchangeable. They are asking for an opinion. In this context, you can often leave off the 'about that', as it wold be implied. On the other hand ... In English, "how do you feel?" has a usage which usually relates to your health. Typical answers to this question would be "I feel fine thanks", or "I think I have the flu today". The other question, "What do you feel?" could be either asking an opinion or a different question - about something that you are touching. As with so many phrases, it all depends on context.

Some examples; What do you feel when you read about poverty and war? Answer: I feel sad and helpless. How do you feel today? Answer: I feel better now my cold has gone.