Gemara

(Mishnah): The two Parshiyos in a Mezuzah are Me'akev each other. Even one Kesav is Me'akev.

28a (Mishnah): The four Parshiyos in Tefilin are Me'akev each other. Even one Kesav is Me'akev.

Berachos 13a (Mishnah - R. Yehoshua ben Korchah): We say Shma before v'Hayah (Im Shamo'a) in order to accept Ol (the yoke of) Malchus Shamayim before Ol Mitzvos;

We say v'Hayah before va'Yomer, because the former (discusses learning Torah, which) applies during the day and night, whereas va'Yomer (discusses Tzitzis, which) applies only during the day.

13b (Beraisa): "V'Hayu" teaches that one must recite in in order.

14b (Beraisa - R. Shimon): It is proper that Parshas Shma precede v'Hayah, for Shma discusses learning, and v'Hayah discusses teaching;

It is proper that v'Hayah precede va'Yomer, for v'Hayah discusses teaching, and va'Yomer discusses performing (the Mitzvos).

Objection #1: Shma discusses teaching and performing in addition to learning -- "v'Shinantam... u'Kshartam... u'Chsavtam"!

Objection #2: V'Hayah discusses performing in addition to teaching -- "u'Kshartem... u'Chsavtem"!

Correction: Rather, it is proper that Shma precede v'Hayah, for Shma discusses learning, teaching and performing, and v'Hayah discusses only teaching and performing. It is proper that v'Hayah precede va'Yomer, for v'Hayah discusses teaching and performing, and va'Yomer discusses only performing.

Question: R. Yehoshua ben Korchah's reason should suffice!

Answer: Indeed, R. Shimon gives an additional reason. It is proper that Shma precedes v'Hayah, in order to accept Ol Malchus Shamayim before Ol Mitzvos. Also, Shma discusses learning, teaching and performing, and v'Hayah discusses only teaching and performing.

15a (Mishnah): If one said it out of order, he was not Yotzei;

If one erred, he returns to the place where he erred.

Yoma 60a (Mishnah): If Avodos of Yom Kipur were done out of order, he did nothing. If he put the goat's blood before the bull's blood, he returns to sprinkle the goat's blood after the bull's blood.

Rishonim

Rambam (Hilchos Kri'as Shma 2:11): If one read out of order, he was not Yotzei. This refers to the order of the verses. If he read a Parshah (that should be read later) before another, even though he may not do so, I say that he was Yotzei, for they are not adjacent in the Torah.

Rosh and R. Yonah (Berachos 2:12 and 8b DH ha'Korei): The Rambam says that if one read the Parshiyos out of order, he was Yotzei, because they are not adjacent in the Torah. Even though the Mishnah teaches why Shma is before v'Hayah Im Shamo'a, this is l'Chatchilah. It is not Me'akev. R. Yehudah says that one may not interrupt between va'Yomer and Emes v'Yatziv when he reads in order.

R. Yonah: When he reads in order, since he says "Ani Hash-m Elokeichem", he should say 'Emes' right after without any interruption. When he reads out of order and it (the last words of Parshas Tzitzis, i.e. Ani Hash-m Elokeichem) is not next to Emes v'Yatziv, we are not concerned.

Rosh (ibid.): If he knows that he read the Parshah, but he skipped a verse or word. He returns to that verse. A Tosefta (2:4) says so. He reads from that verse to the end of the Parshah. The same applies to Hallel, Megilah and Tefilah.

Tosfos (14b DH Lamah): We should say that Shma precedes v'Hayah because it is earlier in the Torah! Rather, we needed to explain why it is before even Parshas Tzitzis, which is the earliest in the Torah. Rather, we are not concerned for this, because we say that the Torah is not (always) in order. If so, we should have put v'Hayah first, for it is in the plural. (The Tana'im come to explain why we do not.)

Poskim

Shulchan Aruch (YD 288:3): If one wrote a Mezuzah out of order, even if he forgot to write one letter, it is Pasul, and it cannot be fixed. There is no need to teach that it is Pasul if he wrote the latter Parshah before the former.

Taz (1): This is astounding. There is more reason to disqualify when it is out of order within a Parshah! We find that if one read the verses of Kri'as Shma out of order, he was not Yotzei, but if he read a Parshah before another, even though he may not do so, he was Yotzei, for they are not adjacent in the Torah! Also here, they are not adjacent to each other! The Rambam saw a need to write that if he wrote the latter Parshah before the former it is Pasul, and Tosfos holds that it is Kosher!

Gra (5): Tosfos (32a DH Dilma) holds that if the Parshiyos were written out of order, it is Kosher. The Rosh brings a Mechilta that disqualifies.

Shulchan Aruch (OC 64:1): If one read out of order, he was not Yotzei. This refers to the order of the verses. If he read a Parshah before another, even though he may not do so, he was Yotzei, for they are not adjacent in the Torah.

Bach (1): The order of the verses is Me'akev, because each Parshah is written in order in the Torah. The order of the Parshiyos is not Me'akev, because the Parshiyos are not adjacent in the Torah.

Gra (1): The Parshiyos are not adjacent in the Torah, i.e. it is only an enactment mid'Rabanan to read them in this order.

Mishnah Berurah (1): "V'Hayu" teaches that one must recite the words like they are (in order, in the Torah). When Chachamim enacted to recite also Parshas Tzitzis, for the Mitzvah to remember Yetzi'as Mitzrayim, they enacted like this (in order).

Mishnah Berurah (2): All the more so he was not Yotzei if he said the words in a verse out of order.

Bi'ur Halachah (DH ba'Meh): If one read the verse "v'Hayu..." before "v'Ahavta", he reads just v'Hayu (and onwards). He does not repeat v'Ahavta. Even though he read it out of order, after v'Hayu, now (that we invalidated v'Hayu,) it is in order. A proof is from Yoma 60a. The Mishnah says that if Avodos of Yom Kipur were done out of order, he did nothing. If he put the goat's blood before the bull's blood, he returns to sprinkle the goat's blood after the bull's blood (but he need not sprinkle the bull's blood again). Based on this, if we would say that only the first verse is mid'Oraisa, the Mi'ut "v'Hayu" (which teaches that one must recite in order) could refer only to saying the words in the verse out of order. Saying the verse of Shma at the end of the Parshah is no worse than saying only this verse.

Gra (on Sa'if 2): A Tosefta (2:4) says that if one read out of order, he was not Yotzei. The same applies to Hallel, Tefilah and Megilah. If one read Shma, erred and omitted a verse, he does not return to read just the verse. Rather, he begins from that verse and finishes until the end. The same applies to Hallel, Tefilah and Megilah. If one entered a Beis ha'Keneses and found that they already read half (of Shma), and he finished with them, he does not go back to read from the beginning until the place where he entered. Rather, he begins from the beginning and finishes until the end. The same applies to Hallel, Tefilah and Megilah. It seems that this is unlike the Rambam. We say that if he omitted a verse, or entered in the middle..., which connotes even in the first Parshah. Even so, he must finish until the end. (Even though he read the last two Parshiyos totally and properly, he must repeat them, for he recited them before he was Yotzei saying Parshas Shma!) We compare this to Tefilah, Hallel and Megilah.

Bi'ur Halachah (DH Af): The Gra learns from the Tosefta that mid'Rabanan, he was not Yotzei in any case of reading out of order. Also the Pri Megadim suggested this. The Pri Megadim asked why the Rambam and Shulchan Aruch disqualify (a Mezuzah) when the Parshiyos were written out of order. Also there, the Parshiyos are not adjacent to each other! The Taz in YD asked like this. I answer that the Gra means that since the Torah wrote v'Hayah after Shma, if one writes v'Hayah before Shma, this is not k'Havayasan (like they are), and it is Pasul. Here (for Kri'as Shma), it is only an enactment mid'Rabanan. This is because the four Parshiyos in Tefilin and the two in Mezuzah are Me'akev each other. If one verse or Parshah is missing, it is as if he did not write at all. Therefore, when the Torah taught that he must write in order, like it says in the Mechilta (regarding a Mezuzah, and presumably the same applies to v'Hayu l'Totafos Bein Einecha), it is like one Parshah. It is divided only into verses. Here, if one read only the first Parshah, in any case he has the Mitzvah of Kabalas Ol Malchus Shamayim, even if he did not read the second Parshah of Kabalas Ol Mitzvos. Even if we will say that both are mid'Oraisa, they are not Me'akev each other. Each has a verse commanding to say that Parshah ("v'Hayu ha'Devarim ha'Eleh... Al Levavecha" in the first Parshah, and "v'Samtem Es Devarai Eleh Al Levavechem v'Al Nafshechem" in the second Parshah - PF). Each Parshah has its own theme. Therefore, v'Hayu teaches only about the verses in each Parshah (one must say them in order. It does not command to say the Parshiyos in order). This is even if both are mid'Oraisa, and all the more so if only the first Parshah is mid'Oraisa, v'Hayu refers only to the order of the verses in that Parshah. In any case, mid'Rabanan surely one is not Yotzei if he read the verses in the second or third Parshah out of order.

Kaf ha'Chayim (3): Based on Kabalah, there is a reason why the Parshiyos must be read in order. Ru'ach Chayim says that if one read out of order, he reads again.