Yes and No
By the standards of the Diagnostic and Statistical Manual of Mental Disorder (or DSM-IV in its current form), perhaps the most prominent all-in-one manual to assist physicians in accurately defining a patient's disorder, has specific criteria for a disorder, including:
- is associated with present distress (e.g., a painful symptom) or disability (i.e., impairment in one or more important areas of functioning) or with an increased risk of suffering death, pain, disability, or an important loss of freedom
A level of empathy such that you've described could certainly fit this definition. In practice most physicians don't consider culturally/socially sanctioned actions as symptoms of mental illness (whether they are is an age-old debate for philosophers, anthropologists, and sociologists/psychologists). Indeed, even the DSM-V has proposed changing criteria C (which rules out acceptable responses to loss/stress) to exclude "a culturally sanctioned response to a particular event." It seems that most cultures value empathy enough that it's unlikely to be viewed as a disorder. In fact, even the most extreme examples of empathy (complete self sacrifice) are often considered admirable things in books and movies.
If, however, empathy is directed toward someone who abuses the relationship, it could result in codependency. While the DSM doesn't list codependency, it is still often diagnosed as a mental illness. In the case of codependency, the empathetic person often cannot bring themselves to end the relationship, due to their concern over the negative effect it would have on the other person. Since this relationship is dangerous to the empath and caused by said empathy, this could be a case where too much empathy in an individual is a mental disorder.
Then again, if the other person had more empathy, the situation wouldn't occur. I think it's safe to say that in theory empathy shouldn't be a disorder but in practice it can lead to disordered behavior.