Gemara: It is no’ach [literally, comfortable, but in context, preferable - see later on] for a person to throw himself into a furnace than to whiten the face of a person (with embarrassment) in public. From where do we know this? From Tamar [who was willing to be burnt rather than expose Yehuda’s impropriety].

Ein Ayah: Just like pursuing imaginary respect is one of the bad traits that make a person lose his place in the world, so too the true recognition of a life of human dignity and the despising of a life of disgrace is a foundation of the world. This elevates a person’s spirit to recognize the true dignity of shleimut (completeness) and the value of truth in wisdom and knowledge of Hashem. Therefore, a person should realize that the value of life stands on its own only when it is connected to the feeling of human dignity. When a person is missing that feeling of dignity, his life is not considered a human life.
It is true that there are people "of a great heart" who are willing to accept degradation with love because they see it as a preparation for a more complete and greater honor, which is the honor of shleimut. However, this is not a sign that human dignity does not have value, for indeed it is the form of human life.
It is based on the above that the gemara says that it is preferable for a person to give up his physical life, even in an unusual manner that is the opposite of dignity for a moment, than to embarrass his counterpart in public in a manner that shrouds his friend’s dignity over the future in an inerasable cloud. This is because the impact of something done in front of many people is very great, and when one loses his human dignity in that forum, he loses his standing and the value of his life.
Therefore, due to the moral aspirations of a complete person who knows how to value life, it is more fitting to prefer losing his physical life than to cause his friend to lose his dignity, which is like an ongoing, undesirable death. It is possible that based on the rules of the Torah, one should not actually give his life under such circumstances because a full life has a great advantage in that his friend can have his dignity that escaped him healed and restored to him. However from the perspective of what one should feel, he should be willing to give his life. One’s thirst for life should give way, and he should view the temporary pain associated with giving his life as nothing compared to the knowledge that someone’s dignity will be trampled publicly, as the latter’s pain will be ongoing and long-lived. That is why the word no’ach [literally, comfortable, but in context it is more of an emotional preference], as this is not an operative halacha but is the way one should feel.
We need to understand the extent of the "whitening of the face" which is being discussed, as there are different levels of embarrassment. It is apparently talking about a case where the shame lowers the perceived value of the person who is the subject of the embarrassment forever in the eyes of the public in a manner that he cannot easily escape it. Only then does it justify giving one’s life even on a theoretical basis.