It looks like telling people "everyone is biased" might make people not want to change their behavior to overcome their biases:

In initial experiments, participants were simply asked to rate a particular group, such as women, on a series of stereotypical characteristics, which for women were: warm, family-oriented and (less) career-focused. Beforehand, half of the participants were told that "the vast majority of people have stereotypical preconceptions." Compared to those given no messages, these participants produced more stereotypical ratings, whether about women, older people or the obese.

Another experiment used a richer measure of stereotyping – the amount of clichés used by participants in their written account of an older person’s typical day. This time, those participants warned before writing that “Everyone Stereotypes” were more biased in their writings than those given no message; in contrast, those told that stereotyping was very rare were the least clichéd of all. Another experiment even showed that hearing the “Everyone Stereotypes” message led men to negotiate more aggressively with women, resulting in poorer outcomes for the women.

The authors suggest that telling participants that everyone is biased makes being biased seem like not much of a big deal. If everyone is doing it, then it's not wrong for me to do it as well. However, it looks like the solution to the problem presented here is to give a little white lie that will prompt people to overcome their biases:

A further experiment suggests a possible solution. In line with the other studies, men given the "Everyone Stereotypes" message were less likely to hire a hypothetical female job candidate who was assertive in arguing for higher compensation. But other men told that everyone tries to overcome their stereotypes were fairer than those who received no information at all. The participants were adjusting their behaviour to fit the group norms, but this time in a virtuous direction.