In introducing some of the details concerning the Korban Chatas, the sin offering, the Torah uses the expression "זאת תורת החטאת," "this is the law of the sin offering" (ויקרא ו':י"ח). Similarly, the Torah a few Pesukim later uses the expression "וזאת תורת האשם," "and this is the law of the guilt offering," to introduce its presentation of some of the details relating to that Korban (שם ז':א'). The Gemara in Menachos (דף ק"י.), apparently focusing on the word "תורת," literally meaning "the law of," in those phrases, but understood to be a reference to "Torah" in the more general sense, derives from those Pesukim (שם ושם) the idea that anyone who involves himself in the study of the Torah, the law, of the Korban Chatas is considered as if he actually has brought a Korban Chatas, and that likewise regarding the Korban Asham, anyone who involves himself in studying the Torah of the Korban Asham is viewed as if he has actually offered a Korban Asham. A similar idea is found in the Midrash Tanchuma in our Parsha (אות י"ד), which stresses that particularly nowadays, when there is no Beis HaMikdash, and thus no Korbanos are brought, one should study and read about a specific Korban, and thus receive the credit for having brought that Korban, thereby guaranteeing himself a reward in Olam Habo.
Rashi in Menachos (שם בד"ה בתורת) understands the Gemara there (שם) as referring to the study of those Halachos relating to the Avodah, the service in the Beis HaMikdash, meaning that one who studies the Halachos about the Korbanos mentioned there is considered as if he actually offered those Korbanos; this also appears to be the understanding of the Maharsha there (חדושי אגדות למנחות שם בד"ה מ"ד), who writes that the author of this view is coming to stress that one who engages in Talmud Torah specifically about a certain Korban is credited as though he actually brought that particular Korban. Rabbeinu Bechaya, commenting on a Posuk later in our Parsha (ויקרא שם פסוק ל"ז), notes that in order to be credited with having offered a particular Korban, one must study the details concerning that Korban in depth, with an eye towards comprehending its true meaning and symbolism, as opposed to merely reading the Torah's words about the Korban without any understanding at all, and one will thereby be inspired to be more careful to fulfill the dictates of the Torah and the Mitzvos and will thus be forgiven for his sins. The Rambam, in his Peirush HaMishnayos in Menachos (פרק י"ג משנה י"א), writes that when Talmidei Chachomim are involved in studying the laws of the Avodah, it is considered as if the Beis HaMikdash has been rebuilt, and it is thus proper to involve oneself in matters related to the Korbanos, and not have the common attitude that these matters are irrelevant nowadays. We thus see the significance of studying the laws relating to the Korbanos even today, and perhaps especially today, in the absence of the Beis HaMikdash.
A similar, although not identical, idea emerges from an Aggadic passage found in the Gemara in Megillah (דף ל"א:) which records a discussion between Hashem and Avraham Avinu based upon the Pesukim which describe the Bris Bein HaBesarim, the covenant during which Hashem promised Avraham that his descendants would eventually inherit Eretz Yisrael (בראשית ט"ו:ז'-כ"א). After hearing this promise, Avraham asked Hashem "במה אדע כי אירשנה," "how can I know that I will inherit it" (שם פסוק ח'). The Gemara (שם, ועיין ברש"י שם בד"ה במה) explains that Avraham was concerned that his descendants might sin and thus be punished, thereby, in effect, nullifying this promise about the inheritance of Eretz Yisrael, if there is nothing to atone for their sins. In a similar vein, the Midrash in Bereishis Rabbah (פרשה מ"ד סימן י"ז) states that Avraham was really asking in what merit (באיזו זכות) he and his descendants would inherit the land; this is the interpretation presented by Rashi, in his commentary on the Torah (לפסוק ו' שם בד"ה ויחשבה). Hashem then answered Avraham by instructing him to set aside certain animals and birds (שם פסוק ט'), which the Gemara (שם), as explained by Rashi (שם בד"ה קחה), and the Midrash (שם) understand as a reference to the Korbanos, meaning that the vehicle through which Avraham Avinu's descendants can be assured of inheriting, and maintaining control over, Eretz Yisrael is the offering of the Korbanos. The Gemara (שם) then goes on to say that Avraham, prophetically anticipating an era when there would be no Beis HaMikdash, and hence, no Korbanos, asked what merit his descendants would have at that time. To this, Hashem responded that He has already established an "order" of the laws and details concerning Korbanos, and that when Bnai Yisrael will read about the Korbanos, He will consider it as if they have offered these Korbanos, and their sins will be forgiven. They will thus be entitled to Eretz Yisrael in that merit. This passage appears as well in the Gemara in Taanis (דף כ"ז:); the Maharsha there (חידושי אגדות שם בד"ה כבר) quotes an opinion that the very survival of the Jewish people in exile depends to some extent on their reading about the Korbanos. Rabbeinu Bechaya, in his commentary cited above (ויקרא שם), notes that it is because of all of this that Chazal incorporated the recitation of certain passages relating to Korbanos into our daily morning davening.
The Tur (אורח חיים סימן א') writes that it is proper to recite the sections of the Torah dealing with the various Korbanos in the morning some time after daybreak; the Beis Yosef (שם בד"ה ופרשת) cites the aforementioned passages in the Gemara in Taanis (שם) and in Megillah (שם) and in Menachos (שם) as the sources for this practice, and the Perishah (שם אות ט"ז) says something similar, as does the Mishnah Berurah (שם סוף ס"ק י"ג). The Shulchan Aruch (שם סעיף ה') also rules that one should recite these sections of the Torah each morning; the Mishnah Berurah, in his Biur Halacha (שם בד"ה ופ' עולה), elaborates as to exactly which sections should be recited, referring to all the different Korbanos. These different sections are popularly referred to collectively simply as "Korbanos," and the Shulchan Aruch (שם סעיפים ח',ט') adds that there are certain other Pesukim and sections that are customarily recited along with the Korbanos. The Mishnah Berurah (שם ס"ק י"ט) notes that our practice today is to recite these things in a slightly different order, and the Aruch HaShulchan (שם סעיף כ"ג) details exactly what the current practice is regarding the recitation of Korbanos. It is noteworthy that the Tur (שם) suggests that a certain text, requesting that Hashem accept what we have done, be recited after describing each of the different Korbanos, and the Shulchan Aruch (שם סעיף ז') rules accordingly. The Bach (שם בד"ה ומ"ש ואחר) indicates that the recitation of this text is what is in place of the offering of the particular Korban, and the Aruch HaShulchan (שם סעיף כ"ח) records the current practice regarding the recitation of this text. The Mishnah Berurah (שם ס"ק י"ז) notes, however, that this text should not be recited on Shabbos or Yom Tov.
It should be pointed out that the Kaf HaChaim (שם אות ל"ו) notes that many people are not careful to recite the Korbanos every day, but he adds (שם) that at times of need, it is especially propitious to do so, and the Magen Avraham (שם ס"ק ז', ועיין שם ביד אפרים בד"ה כתב הבחיי) as well as the Mishnah Berurah (שם ס"ק י"ג) stress that it is not enough to simply read the Pesukim about the Korbanos, but one should rather examine and understand what he is saying in order to recognize the greatness of Hashem, as mentioned by the Rabbeinu Bechaya in his above cited commentary (שם). In the aforementioned Biur Halacha (שם), the Mishnah Berurah adds that reciting the Korbanos is a significant Mitzvah, and that one should first try to learn about these matters from the Gemara or the Rambam in order to understand them and thereby be given credit as if he has actually offered a Korban. We can thus see the significance placed by these Poskim on reciting the Korbanos every day, and, as noted above, the Jewish people's right to Eretz Yisrael is in some way dependent on this recitation.
One of the sections of the Torah which appears to be given prominence in terms of the requirement to recite it as part of the Korbanos section of the davening is the Parshas HaTamid, the section describing the daily offering in the Beis HaMikdash (במדבר כ"ח:א'-ח'). The Shibolei HaLekket (סימן ה') writes that reading the Parshas HaTamid is an obligation, particularly because no reference to it is made in the Shemoneh Esrei; this is unlike the various Pesukim regarding the Korban Mussaf (for Shabbos, Yom Tov and Rosh Chodesh) which are mentioned in the Mussaf Shemoneh Esrei. The Kol Bo (סימן ב') adds that the practice is to recite the Parshas HaTamid daily because we do not read those Pesukim in the Torah every day (as we do the Pesukim about the Korban Mussaf on Yomim Tovim, for example); the Sefer HaEshkol (חלק א' סימן ד') expresses a similar idea, and the Beis Yosef, commenting on the Tur later (או"ח סימן מ"ח בד"ה בעל), writes that this is indeed the custom. The Tur (שם) lists several other Pesukim and passages which ought to be recited in conjunction with the Parshas HaTamid, and then writes that it is proper to recite a small, personal Tefillah after all this, primarily to ask that Hashem should accept the reading of all these passages as though the Korban Tamid has indeed been offered properly; in the Shulchan Aruch (שם סעיף א'), the Ramo appears to concur. The Kaf HaChaim (שם אות א') describes at length exactly what is to be said when reciting the Korbanos section of the davening; the Mishnah Berurah (שם ס"ק א'), after outlining the basis for reciting Korbanos altogether, appears to say the same thing, and he adds that one who knows how should learn about the meaning of all these passages in order to understand what he is saying, and not just utter the words.
The Mishnah Berurah, in the Biur Halacha cited above (או"ח סימן א' בד"ה ופ' עולה), writes that it is proper to recite the Korbanos specifically in Shul, possibly because every Shul, in a certain sense, is a scaled down replica of the Beis HaMikdash, where the Korbanos were actually offered, as indicated by the Gemara in Megillah (דף כ"ט.). The Magen Avraham (שם סימן מ"ח בהקדמה לסימן) writes that one should be standing when reciting the Korbanos, just as one who actually offered a Korban would stand, and the Be'er Heitev, both here (שם ס"ק א') and earlier (שם סימן א' ס"ק י"ב), appears to agree. The Shaarei Teshuvah (שם סימן מ"ח ס"ק א'), however, quotes many who disagree, as does the Kaf HaChaim both here (שם אות ב') and earlier (שם סימן א' אות ל"ג). Among them is Rav Yaakov Emden (ספר מור וקציעה שם סימן מ"ח), who reports that his father the Chacham Tzvi did not insist on standing when reciting the Korbanos, and he explains that reciting the Korbanos daily is really a Minhag, since there is no hint to such a formal practice in the Gemara, and one therefore need not be so strict in terms of modeling the recitation of the Korbanos after the actual offering of the Korbanos, because this recitation is merely a commemoration of what was done in the Beis HaMikdash. In his commentary to the Siddur, however, Rav Yaakov Emden writes (סידור בית יעקב, עזרת ישראל, בתחילת ביאורו לפרשיות הקרבנות) that there may be Kabballistic reasons for standing when reciting Korbanos in the morning. The Pri Megadim (באשל אברהם לסימן מ"ח שם בהקדמה לסימן) also quotes those who hold that it is not necessary to stand when reciting Korbanos, although he adds (שם) that it is proper to stand when reciting the Parshas HaTamid. The Aruch HaShulchan (שם סימן א' סעיף כ"ו) proposes the interesting idea that a Kohein should stand when reciting the Korbanos because he would actually be offering the Korban (while standing) were the Beis HaMikdash still here, but a non-Kohein may sit when reciting Korbanos. It should be noted that according to the Mishnah Berurah in his Biur Halacha (שם סימן מ"ז בד"ה נשים), women also are obligated to recite Korbanos; this is also the view of the Magen Avraham (שם ס"ק י"ד), but others, including the Pri Megadim (באשל אברהם שם ס"ק י"ד) and Rav Yaakov Emden (ספר מור וקציעה שם סוף סימן מ"ז), disagree. It should also be noted that the Ramo (שם סימן רל"ד סעיף א') rules that it is proper to recite the Parshas HaTamid in the afternoon before Minchah as well.