This topic has been met with recent controversy. Specifically, several Asian-American students have been very reluctant to identify their race in fear of being held to higher academic standards than their peers (often referred as reverse discrimination). This hones in on the issue that colleges and universities have an applicant quota for races and ethnicities and applicants within their racial groups are competing against one another. On one hand, some believe that you are not revealing who you are as a person by not disclosing your race, which is what most colleges typically preach that they want. Conversely, some feel that racial identification intensifies admission competition for them and shrinks their chance of acceptance. The key question is 'where does college draw the line between creating a diverse academic community and discrimination?'
Overall, there is no benefit to check 'no race' on your application. If a college does not accept you based solely on your racial identity, it is not the right college for you, regardless of how prestigious it may be. There are many proxies that the admission team can use other than race to build a diverse intellectual community.