Is the obligation to answer amen a biblical or rabbinic mitzvah? If it is a biblical mitzvah what is the source of the obligation?
According to most poskim, there is a rabbinic obligation to answer amen upon hearing somebody else recite a blessing. Chazal in several places highlight the importance of answering amen, and several poskim (mainly based on kabbalistic sources) mention the failure to answer amen as being a serious misdeed.
Sources: Mishnah, Berachos 51b; Rambam, Laws of Blessing 1:13 (writes that there is an obligation to answer amen, unlike Tur, Orach Chaim 215, who writes that there is no obligation). Concerning the importance of answering amen, see Berachos 53b; Shabbos 119b; Tanchuma (Parshas Tzav); Midrash Rabbah (Devarim 7); Sefer Chassidim (863). Concerning the stringency of failure to answer amen, see Kaf Hachaim (56:9, 124:30); Zohar (Bereishis 285a); a famous story is recounted of the punishment given to somebody who failed to answer amen in Nahara Hashalem, p. 15.