The famous Halachic principle of "Tashlumin" allows a person to make up a missed Amida prayer if the prayer was missed inadvertently. For example, if a person forgot to recite the Shaharit prayer, he recites an extra Amida at Minha to make up for the missed prayer. Likewise, if a person forgot to recite Minha, he recites an extra Amida at Arbit, and an inadvertently missed Arbit prayer is made up the next morning, during Shaharit.
Does this apply to Shema, as well? If, for example, a person fell asleep early one night before reciting Arbit, and he thus missed both Shema and the Amida, should he recite an extra Shema the next morning (in addition to an extra Amida) to make up the missed Shema recitation?
Although this might initially sound like a peculiar question, this issue is actually subject to debate among the Rishonim (Medieval Halachic authorities). The Shulhan Aruch, in Orah Haim (58:7; listen to audio recording for precise citation), cites two different views. The first view maintains that one can, in fact, make up a missed Shema recitation. According to this view, somebody who forgot to recite Shema at night should recite an additional Shema the following morning, after the Amida during Shaharit, and a person who forgot to recite Shema in the morning should recite an additional Shema that night, after Arbit. This was the view of a number of prominent Rishonim (the Roke’ah, Orhot Haim and Re’avya). The second view cited by the Shulhan Aruch, which is the majority opinion, maintains that the concept of "Tashlumin" applies only to the Amida prayer, and not to Shema. The Amida prayer is an appeal for divine compassion, and Halacha allows a person who missed one prayer to bring his appeal to the Almighty on a different occasion to make up for the missed prayer. Beseeching God is not something which is inherently restricted to one time frame or another, and so a person who missed a prayer has the opportunity to make it up. The Shema obligation, however, is a specific requirement that applies "when you lie down and when you arise"; we bear an obligation to recite the Shema each morning and each evening. If a person misses the evening obligation, then, he cannot make it up the next morning, since the obligation is inherently limited to the evening hours.
The Hid"a (Rav Haim Yosef David Azulai, 1724-1806) follows the second opinion, which does not allow for making up a missed recitation of Shema, and this is, indeed, the accepted Halacha. The Hid"a applies to this case the verse in Kohelet (1:15), "Me’uvat Lo Yuchal Litkon" – "Something distorted cannot be fixed." Once a person missed the evening or morning Shema recitation, there is no possibility of rectifying the mistake and making up the missed recitation.
Summary: As opposed to the Amida prayer, which one can make up in a case where he inadvertently missed a prayer, there is no possibility of making up a missed Shema recitation.