Empathy is an invaluable skill for a leader - I would argue that it should be one of the core skills that anyone wanting to be a manager should have. It is also important to remember that empathy can have some consequences of which you need to be aware.
A recent study shows that leaders with high empathy give more effective feedback, but also struggle with their performance after giving negative feedback. The belief is that because they are so focused on providing effective feedback and caring about the other person that it can cause a drain on them after giving it. The best way I think about it is that it drains the gas tank of the empathetic leader.
This helps because empathetic leaders should be aware of this. By knowing it happens, they can plan properly for it. Maybe they delegate more knowing they won't be at their best. Maybe they can plan negative feedback for when they will have a chance to recover afterwards.
The lesson: Managers need to understand how various responsibilities affect them and make the necessary adjustments. No one, not even the greatest manager, is great all the time.
NOTE: the lesson: this doesn't mean be any less empathetic. This is a time when something that causes a drop in performance (temporarily) is just fine.