I don't think mixed methods is any different from either qualitative or quantitative methods with regard to bias. Instead, each component of a mixed methods design has to follow best practices for dealing with design in that particular method.

I believe that researcher bias may occur more in qualitative research than quantitative because qualitative can be more subjective in nature. Hence, in mixed method research where you are mixing both quantitative and qualitative components, the quantitative analysis can bring about more objectivity and could serve as neutral validation for the study and possibly minimise/mitigate subjectivity of the qualitative.