Why does the Torah command us to kill a navi sheker (false prophet)? Shouldn't we attempt to find help for the fellow by directing him to seek psychiatric help?
See Sefer Hachinuch 517:
כי האיש הזה באמרו שהוא נביא ונצטוה בזה, ונראה דבריו מתקימין כדברי נביאי האמת, נחזיק אותו כאיש אלקים, קדוש, שליח האל, ונאמין אליו ונקח ראיה בכל הנהגותינו ממעשיו, ואולי אחר שאין זכותו וענינו גדול להיות הוא שליח באותה נבואה שאמר איננו ראוי לסמך בו בכל אשר יעשה ויאמר, ויטעה ההמון בלכתם אחר עצתו.
People may err and believe that this man is near to Hashem, believing him and learning from him.
"Forensic psychiatry speaking", the Sanhedrin never put to death someone who was not in correct mental state at the time of offense or someone with no competency to stand trial. If he believes by mistake that he is a true prophet, he would not be killed. To kill someone you need to know that he is mezid and sane. Insane is called Shote. If he is not Shote but is doing a mistake he is Shogeg st least.
In this particular case, we are told explicitly in the Torah of Moshe, Devarim 13:1-6, that such a person is sent directly by G-d (they are moshiach HaShem, one of G-d's anointed, so to speak) for the purpose of testing us to see if we love HaShem with all our heart and soul and follow Him alone, fulfilling His commandments and not transgressing them.
Sforno explains that the goal of the false prophet is to convince the Jewish people to stop following G-d's commandments. That is what is meant by דבר-סרה mentioned in the sixth posuk.
And Moshe tells us that the way that we demonstrate that we have passed His test is not to listen to the false prophet, nor to follow his directives and to remove him (execute him) and remove his teachings from our midst.
In other words, it is an actual mitzvah to execute such a person, just like it is a mitzvah to put on tefillin or not to stand idly by when your brothers blood is being shed.