Accurate utterance classification in motivational interviews is crucial to automatically understand the quality and dynamics of client-therapist interaction, and it can serve as a key input for systems mediating such interactions. Motivational interviews exhibit three important characteristics. First, there are two distinct roles, namely client and therapist. Second, they are often highly emotionally charged, which can be expressed both in text and in prosody. Finally, context is of central importance to classify any given utterance. Previous works did not adequately incorporate all of these characteristics into utterance classification approaches for mental health dialogues. In contrast, we present M3TCM, a Multi-modal, Multi-task Context Model for utterance classification. Our approach for the first time employs multi-task learning to effectively model both joint and individual components of therapist and client behaviour. Furthermore, M3TCM integrates information from the text and speech modality as well as the conversation context. With our novel approach, we outperform the state of the art for utterance classification on the recently introduced AnnoMI dataset with a relative improvement of 20% for the client- and by 15% for therapist utterance classification. In extensive ablation studies, we quantify the improvement resulting from each contribution.