Feeling about refers to when you talk of your feelings related to someone or something.
For example:
1) I feel angry about students doing politics in college.
2) I am not getting a good feeling about this.
3) What are your feelings about the growing trend of live-in relationships?
Feeling for means sympathize. Its like ‘feeling someone else’s pain’, for example:
1) Sudha has no feelings for her violent ex-husband now.
2) I do not feel for people who do not work hard and then suffer.
3) Why do you feel so much for everyone?