"Psychiatric Illness" can mean something so serious that the potential donor does not really understand what he/she is doing. For example:
"The little green men from the space ship told me to donate a kidney" or "my left kidney is possessed by a demon, so must be removed". Or it could be a condition that makes it difficult for the patient to make a free and well-informed decision, for example a person struggling with many mental or personal problems who unrealistically thinks "if I donate a kidney it wil be such a nice thing that it will make all my problems go away and make everyone love me."
Persons under treatment/medication for depression also should know that some donors suffer post-op depression, even if the donation is a success. So they have to consider their ability to handle an extra dose of depression if it should hapen.
And of course the hospital must be aware of ALL medications the potential donor is taking, to avoid bad inter-actions with the anesthesia, pain-killers, etc. that will accompany the surgery.
So, just be completely honest with the transplant coordinator. They are not automatically ruling out everyone under some psychiatric treatment/medication. They just have to be sure that the potential donor understands and freely accepts what she/he is doing, and shows sufficient signs of being able to handle the psychological pressures that donation can sometimes involve.
best wishes,
fr. Pat