People who are apathetic are overall bad for society. They don't contribute to the advancement of ideas or the advancement of humanity. Apathy allows oppression to take place, oppression is immoral. People need to care for anyone to survive or be healthy. You cant raise a child successfully if you are apathetic towards that child. You are taking away from an innocent with your apathy. You are taking away from society with your apathy.
To imply that apathy is immoral would be to say that not caring about any particular issue is immoral. Because apathy is the state of indifference, it is not usually total. You also mentioned that it is immoral because apathetic people don't contribute much to society, but if you really think about it, what have you are nearly anyone on the planet done that matters? Furthermore, there is no responsibility to care about suffering, not caring is not taking a stance, but the lack of a stance, how can apathy be immoral when apathy isn't a stance or an action?
Asking if apathy is immoral is pretty black and white. Apathy about what? Maybe some things are worth being apathetic about, whereas other things it's harmful to not care. It's harmful to be apathetic about the health of the planet, since we all live on it, but it's not harmful to be apathetic about... I don't know... someone's spilled milk.
Apathy is more like a opinion. People have different things to be apathetic about, so how is it possible to decide if it is immoral or not. For example a person is sympathetic on cancer patients but apathetic on starving people.While another person has the opposite opinion.Who has the right to decide which one is immoral?
I don't know if you mean apathy over everything or what, but there all things we just don't give a damn about. Not caring about your own child, is bad, not caring over if a historic building get torn down or not, doesn't hurt anyone. I could give you a better answer for the question if you worded it better. But obviously you were apathetic about if people would understand you.