Breaking another persons heart is synonymous to hurting them emotionally. The general answer would be yes although there are Shariah exceptions.
If one is unaware that they hurt someone - upon finding out, they need to analyse if the other person has a right to feel heart broken and to what capacity.
Example: Husband tells wife he wants a cup of coffee whilst lazing around watching TV. Wife gets so busy in kitchen that she forgets. Husband claims he is heart broken. In this case, the wife is only partially responsible for forgetting - NOT breaking his heart... If he really needed the coffee, he could have helped himself. His claim here to be "heart broken" is just a sympathy ploy and frankly pathetic. In other words - because someone claims to be heart broken doesn't mean they have a right to be heart broken.
If someone breaks another persons heart, he/she recognizes that, he/she understands the shariah perspective is that he/she was wrong in breaking the other persons heart, then an apology has to be rendered.
In certain circumstances in Islam - it is permissible : Example: A Muslim man in love with Non-Muslim. Relationship is understood as being in love. He tries his best Dawah to bring her into Islam with no avail yet is involved in Zina with her concurrently. His realization is he has to come out of the sin and tried his best to bring her to Islam unsuccessfully. He now decides to move on and be closer to the practice of Islam and leaves her. Technically he has broke her heart but it served him glad tidings as he is now out of sin.
Very long topic with loads of permutations and circumstances. I'm sure the others will join in with much more to ponder about.