The gemara cites Rava who learns from the pasuk of העולה that the korban Tamid has to be the first one brought in the day, and learns from והקטיר עליה חלבי השלמים that nothing can come after the evening korban Tamid, rather עליה השלם כל הקרבנות כולן. Tosafos asks that there is another source in perek ha'tadir that the gemara learns from מלבד עולת הבוקר אשר לעולת התמיד, implying that the Tamid is first. Why do we need both pesukim? Tosafos rejects the possibility that repeating it is to tell you it is me'akeiv, meaning that a korban would be invalid if brought afterward, because the gemara says explicitly that it is not l'ikuva. In Tosafos second answer they say that the pasuk in our gemara focuses on the burning of אימורין on the mizbeiach prior to the Tamid, whereas the pasuk in hatadir is to tell you that the shechita of the Tamid is first. Tosafos also points out that the gemara in Eiruvin says that a korban shelamim brought prior to the opening of the doors of the heichal is invalid, proves that shechting before the tamid won't invalidate, because if it were then we wouldn't need a pasuk to invalidate prior to the opening of the doors.
Tosafos cites the gemara that says that a korban brought prior to the morning tamid wouldn't become passul, but isn't clear about whether a korban that is brought after the korban tamid would become passul. Tosafos in menachos writes that it wouldn't become passul if brought after the evening tamid. Tosafos on thid daf implies that as well because the Tosefta says that a korban brought prior to the morning tamid or after the evening one would become passul, to which Tosafos writes that it is only m'drabonon, implying that they are the same and that m'doraysa neither would be passul. Rashi d.h. kach, writes that nedarim and nedavos brought after the evening Tamid would become passul. See Mishneh L'melech (1:3 of Hil. Temidin U'musafin who is ma'arich very much on these issues).
The Mishna L'melech points out that the Rambam seems to hold like Tosafos because the Rambam (1:3) writes it in the form of an issur, but makes not mention of korbanos being invalid if brought before the morning tamid or after the evening one.
The Rambam writes - לפי שאסור להקריב קרבן כלל קודם תמיד של שחר, ולא שוחטין קרבן אחר תמיד של בין הערבים חוץ מקרבן פסח לבדו שאי אפשר שיקריבו כל ישראל פסחיהן בשתי שעות
The Rambam writes that the only korban brought after the evening Tamid was the korban pesach and the rationale he offers is that if you were to bring the korban pesach prior to the tamid, it would need to be brought between 12:30pm and 2:30pm and two hours isn't sufficient time for the korban pesach. Therefore, it is brought after the Tamid. The Lechem Mishna points out that this rationale is difficult because if the problem were merely the limitation of 2 hours they should have allowed it to be brought either before or after the Tamid. Furthermore, the gemara gives another rationale, since it says by Pesach both בערב and בין הערבים, it implies that it is pushed to later.
Another difficulty with the Rambam is pointed out by the Mishna l'melech at the end of his esssay. Why does the Rambam say that no korban is "brought" prior to the morning Tamid and no korban is "shechted" after the evening one. Tosafos holds that by the morning Tamid where there are two pesukim, one indicates the issur to be makriv (הקטרת אימורין) and the other indicates the issur to shecht, implying that the pasuk cited in our gemara for not sacrificing after the Tamid would be consistent with the context and forbid הקטרה after the evening Tamid. This seems to be the opposite of the Rambam! The Mishna L'melech explains that technically there is no issur to shecht before or after the Tamid, the entire issur is only on being makriv. The reason is that shechita is not regarded as an Avoda and therefore there are no rules limiting it. In the morning one could shecht prior to the morning korban tamid. But, in the evening since if one were to schecht they would be unable to be burn the korban properly or sprinkle the blood, it is assur to even shecht because the are causing the korban to be wasted. Therefore, the Rambam is very meduyak that before the morning Tamid it is only forbidden to be makriv, whereas after the even Tamid it is even forbidden to shecht.