"Having feelings for" a person is an idiom, which means that you are romantically interested in the person you are speaking about.
You would not say it if you merely sympathized, or understood them, it's only for romantic interest.
It's always used in the plural, in this idiom. That signals that it's about romantic feelings, as in your second use example, and not about empathy, as in the first.
On the other hand, in the first example, the noun is uncountable, and not plural.
This means that a singular form is used to represent a quantity that is not discrete - there is not a number that describes "how many". For example:
Bring a jug with some water in it, please.
This is a singular form. It does not make sense to ask "how many waters" are in the jug. The water is uncountable.
You have no feeling for the sufferings of others.
"Feeling" here is a singular form. It does not make sense to ask "how many feelings for the suffering of others" the listener has. So the feeling is uncountable.