Some do. MGH Institute of Health Professions does not.
I am not sure what importance a white coat holds for you in terms of graduation and/or in practice. In my experience, if I wear a white coat into a patient's room they think I am the doctor and ask questions one would pose to a physician or doctor. For patient's who have mental health and/or cognitive issues, it is very confusing for them to see a white coat and hear you are an OT or PT because they have associated a white coat with a doctor so confusion about your role is present. Just food for thought.