The Mishna (Berachot 2:1) discusses in which circumstances one may interrupt the reading of Keriat Shema and distinguishes between different parts of the Shema. Based on this discussion, the Shulhan Aruch (O.H. 66:1) rules that if one is in between sections of the Shema, one is allowed to greet a distinguished individual and one may respond to the greeting of any person. When one is within a section of the Shema, however, one may only greet a person that one fears, such as one’s father or rabbi, and one may only respond to the greeting of a distinguished person.
It should be noted that in the context of this discussion, Keriat Shema refers to the Shema that is recited with blessings, such as during Shaharit and Arvit. If one is reading only the actual verses of Shema with no blessings, such as if one is repeating it after Tzet HaKochavim, although one should accord it the proper honor and try not to interrupt it, it does not have the same status as Shema that is said with blessings.
Nowadays, this is less applicable since people tend to understand that one is reading Shema and not only forego any honor, but also do not expect one to interrupt one’s prayer. Furthermore, unlike Talmudic times during which people feared Roman generals and the like, nowadays thankfully this is not the case. That said, there are situations in which this may be applicable, such as interrupting Shema for one’s non-Jewish boss, and each case should be dealt with individually
One area which is more applicable, however, is if one needs to recite a blessing or respond to the prayer while reading Shema. The Shulhan Aruch (ibid:2-3) says the if one forgot to don one’s Talit or Tefilin, one may stop in between sections and done them with a blessing. Furthermore, one may respond to Kaddish, Kedusha and Barechu even within a section. One may respond to Modim but may only say the word “Modim”.
Another circumstance is if one needs to recite a blessing which is limited in time, such as if one hears thunder and needs to respond immediately. The Mishna Berura (M.B., O.H. 66:19) records a disagreement in such a case. One opinions says that should not lose the opportunity to recite a time-limited blessing and therefore one may interrupt the Shema. Another opinion says that when one is reading Keriat shema, one is already praising Hashem and therefore it is unnecessary to interrupt the Shema to recite another praise. The HIDA (Kesher Gudal, § 9:19) and the Bet Oved side with the latter opinion and it appears that the Sephardic approach is to not interrupt the Shema even for a fleeting blessing. I
In general, Keriat Shema may be interrupted when someone’s honor is involved, such as if one needs to respond to someone so as not to disrespect or embarrass them, or in order to praise Hashem by responding to certain parts of the prayer. The Kaf Hahaim (K.H., O.H. 66:7) adds that another permitted circumstance would be to prevent someone from performing a sinful act. When it comes to the loss of money however, such as responding to one’s financial advisor while reciting Keriat Shema, it does not appear to permissible to do so, even if the sum is great.
Summary: Generally speaking one need not interrupt Shema out of fear or honor to greet another person. One may interrupt Shema in order to don one’s Talit or Tefilin if one forgot to do so, or to respond to Barechu, Kaddish, Kedusha or the first word of Modim. One should not interrupt the Shema for a fleeting blessing.