In addition to the above excellent points, consider the power of stereotypes to deny a person's individuality, identity, and potential. By using a simple and sweeping generalization to define the characteristics of an entire race or group of people, we deny the very existence of the individual within that group. Stereotypes limit the very people they are used to identify and describe, and do so in a way that requires no acknowledgement on the part of those doing the stereotyping that such groups are, in fact, complex and dynamic, and made up of individuals with strengths, weaknesses and characteristics that cover the entire spectrum of human capability.
Asians can be artists or athletes. African-Americans can be professors or writers. Whites can be musically talented and culturally aware. Stereotypes deny all of these possibilities and realities.
In short, stereotypes damage both the person or group that is stereotyped, and the person or group doing the stereotyping. It limits us all by confining our potential, as well our capacity to recognize that potential, while dividing us at the same time.
To me, the most important way in which stereotypes hurt us is by making us believe bad things about ourselves. Tests have shown that we tend to internalize stereotypes about our "kind" of people and that these affect the way we perform.
The experiment that has most affected my thinking about stereotypes is described in the Time link below. It shows that when students are reminded of a negative stereotype about their kind of people, they tend to perform worse on tests that pertain to that stereotype (in other words, women perform worse on math tests when reminded of the stereotype that women are bad at math, etc). This shows me that the power of stereotypes to harm our performance is immense.
I often wonder (since I am very interested in sports) if this is part of why there are so few white Americans in the NBA. If they internalize the idea that they can't compete with African Americans, it may affect their performances or even affect them by pushing them away from playing basketball at all. I find it very interesting that there are (it seems) more good white foreigners than there are white Americans in the NBA and I wonder if this has to do with stereotyping.