The Shulhan Aruch (Orah Haim 215:2) rules that one may not respond “Amen” to the blessing of a heretic, or of a minor who is unaware to Whom one is blessing. If a child is aware that he or she is reciting a blessing to Hashem and there is a real purpose to that blessing, such as eating food, etc., then one may answer “Amen”.
The Biur Halacha (ibid.) discusses a situation in which one is reciting a blessing that one considers a bona fide blessing, but the one listening considers it a Beracha Levatala, a blessing in vain. In such a case, the Mishna Berura (O.H. 215:10) says that as long as there is a Halachic basis or opinion to support reciting the blessing, the one listening may respond “Amen”. One is example is if one recites a blessing on the abridged Hallel on Rosh Hodesh, as is the Moroccan custom, and is in the presence of someone from another Sephardic background who do not recite this blessing, the latter may respond “Amen”.
Rabbi Ovadia Yosef (Yabia Omer, vol. I, § 29) does not give a blanket authorization to answer “Amen” in such situations, but rather says that each situation has to be dealt with individually. Nevertheless, elsewhere in Yabia Omer (vol. V, Orah Haim, § 18), he writes that as long as there is a Halachic foundation for reciting such a blessing, one may answer “Amen” to it, even if one would not recite such a blessing oneself.
Summary: One may only answer “Amen” to a bona fide blessing. One may answer “Amen” to a blessing that is recited based on a legitimate Halachic opinion, even if one does not follow such an opinion oneself