לרפואת הר' אברהם יוסף בן ר' משה חיים
הר' זאב חיים בן חי' אידל
לע"נ ר' אלכסנדר זושא ב"ר יוסף
ומרת אסתר בת שמואל
Let us say that someone was מחלל שבת for פיקוח נפש, does he need a כפרה?
Bi-pashtus, no! He did a MITZVA. Saving life is a mitzva. Almost as big as eating Yerushalmi kugel on Shabbos.
In fact it is a huuuuuge machlokes rishonim [who in all likelihood never tasted Yerushalmi kugel].
The Ohr Zarua brings both sides. There are many other people as well who discuss this and take sides. The Mishna Brura [334/78] says that no kappara is necessary.
Apparently, this depends on the question of whether Shabbos is הותרה or דחויה. If it is הותרה and we understand הותרה to mean completely permitted as if it were חול, then no כפרה necessary. But if it is דחויה and בקושי we allow Shabbos to be violated [as is the definition regarding Tumah] then כפרה is necessary.
Some sefarim bring a proof from the gemara Shabbos [156a] that says that a child born on Shabbos needs a כפרה because they had to be מחלל שבת in order to help his mother give birth. So we see that כפרה is necessary.
We should note [so we will:-)] that there is another opinion in the gemara there that a child born on Shabbos is HOLLLY because he was born on this holy day. What about the חילול שבת? Maybe according to this opinion there is no aveirah at all when one is מחלל שבת for פיקוח נפש.
I just conducted a search of sefarim and found a sefer called עמוד אש by Rav Elazar Shtrashon [published in 1912 - it seems that the author is the grandson of the רש"ש] who says what I just wrote. ותעלוזנה כליותי.
I wanted to make another suggestion. When the gemara says that כפרה is required for a baby born on Shabbos because of the חילול שבת that was done on his behalf, maybe it refers to חילול שבת that wasn't necessary and done mistakenly because of the tension felt at that critical time. But if it was necessary in order to save his life it is fine.
Maybe - and maybe not.