What are reasons why would a psychiatrist ask you to stick out your tongue?

Physical exam: Psychiatrists are physicians as well. Sometimes they need to do a physical exam to ensure that the patient's psychiatric symptoms are not due to a medical problem.

Ck Psychotropic S-E: ECOGNIZING SIDE EFFECTS OF NEUROLEPTICS Movement symptoms may be so subtle that a psychiatrist or neurologist who specializes in movement disorders may be the only expert to pick them up. But in many unfortunate patients, the symptoms are visible from blocks away. Tongue and facial symptoms are often the first to appear and a thorough neurological exam involves careful observation of the tongue in the mouth and sticking out. EPS and TD can mimic disorders such as Parkinson's Disease, Tourette's Syndrome, Huntington's Chorea, tics, cerebral palsy, stroke and hyperactivity.