Our Gemara on Amud Beis warns of the severity of causing another person public humiliation, to the extent that he or she would turn white: 

מַר רַבִּי שִׁמְעוֹן חֲסִידָא, וְאָמְרִי לָהּ אָמַר רַבִּי יוֹחָנָן מִשּׁוּם רַבִּי שִׁמְעוֹן בֶּן יוֹחַי: נוֹחַ לוֹ לָאָדָם שֶׁיַּפִּיל עַצְמוֹ לְתוֹךְ כִּבְשַׁן הָאֵשׁ, וְאַל יַלְבִּין פְּנֵי חֲבֵירוֹ בָּרַבִּים. מְנָלַן — מִתָּמָר.

Rav Zutra bar Tuviyya says that Rav says, and some say Rav Ḥana bar Bizna says that Rabbi Shimon Ḥasida says, and some say that Rabbi Yoḥanan says in the name of Rabbi Shimon ben Yoḥai: It is more amenable for a person to throw himself into a fiery furnace if faced with the choice of publicly embarrassing another or remaining silent even if it leads to being burned, and not humiliate another in public. From where do we derive this? From Tamar, as she was prepared to be burned if Judah did not confess, rather than humiliate him in public.

Tosafos wonders why this is not included in the three sins that one must martyr himself for (as delineated in Pesachim 25a.)  Regardless of Tosafos’ answer, we see that Tosafos takes this statement literally and halakhically, and not merely as an aggadic hyperbole. Rabbenu Yonah in Shaarei Teshuva (III:137-139) seems to also make the case that a sin that is related in kind to one of those big three are also under the obligation (or at least ethical expectation) of martyrdom.  Thus, he says humiliating someone to the point that they will become white as a ghost, is considered a related sub-violation of murder, as the victim feels like they he is dying and thus also subject to martyrdom.

However, Meiri (Berachos 43b) understands it to be an exaggeration: “דרך צחות אמרו”, and so it would seem to be halakhically.  Rav Asher Weiss wrote about this as well, click here, quoting Rav Yehoshua Leib Diskin, that it was a Middas Chassidus on Tamar’s. Rav Weiss also raises a brilliant question and proof that it must be more of an ethical directive than a halakhic fact. After all, we have learned in Sanhedrin (72b) that one can kill a rodef, that is, one who is trying to kill you, even if the person is acting unintentionally.  The Gemara over there rules that this is one of the bases for allowing an abortion if the mother’s life is in danger. The baby is considered a rodef, even though obviously, it has no intention to cause its mother harm. If so, why would Tamar have to give up her life, after all, Yehuda was trying to kill her! If you can kill someone who is trying to kill you, surely you could embarrass somebody who is trying to kill you. Therefore, you must say that Tamar was acting in an extra legal manner regardless.

The lessons for interpersonal behavior speaks for themselves. In fact, if we additionally consider Rav Asher Weiss’ point, we must consider that the emotional pain of embarrassment is even worse than death itself. Because otherwise, it wouldn’t make much sense for Tamar to make that sacrifice. It would be like spending $1000 of your money to save $200 of your friend’s money. That is not Middas Chassidus, that just plain foolish, because there is a net loss in terms of human suffering.