Research on automated mental health assessment tools has been growing in recent years, often aiming to address the subjectivity and bias that existed in the current clinical practice of the psychiatric evaluation process. Despite the substantial health and economic ramifications, the potential unfairness of those automated tools was understudied and required more attention. In this work, we systematically evaluated the fairness level in a multimodal remote mental health dataset and an assessment system, where we compared the fairness level in race, gender, education level, and age. Demographic parity ratio (DPR) and equalized odds ratio (EOR) of classifiers using different modalities were compared, along with the F1 scores in different demographic groups. Post-training classifier threshold optimization was employed to mitigate the unfairness. No statistically significant unfairness was found in the composition of the dataset. Varying degrees of unfairness were identified among modalities, with no single modality consistently demonstrating better fairness across all demographic variables. Post-training mitigation effectively improved both DPR and EOR metrics at the expense of a decrease in F1 scores. Addressing and mitigating unfairness in these automated tools are essential steps in fostering trust among clinicians, gaining deeper insights into their use cases, and facilitating their appropriate utilization.