Gemara

58b - Beraisa - R. Akiva: The Kohen (after leaving the Kodesh ha'Kodoshim) first puts blood on the southeast corner of the inner Mizbe'ach, then the southwest, northwest, and northeast corners.

Question: Granted, R. Akiva does not learn Avodas Panim from Chutz (to require turning to the right). But why does he require turning to the left?

Answer: Really, he should begin with the southwest corner, for he encounters it first, and Reish Lakish taught that Ein Ma'avirin (we do not pass) over Mitzvos;

He does not begin with it, for it says "V'Yotza El ha'Mizbe'ach" - the entire Mizbe'ach (he must reach the end before putting blood on it);

After reaching the east end (of the south side) and putting blood there, he returns to the (southwest) corner where he should have put before.

68b - Mishnah: One who sees the Kohen Gadol read the Torah does not see the burning of the Par and Sa'ir. It is not forbidden. Rather, they are far apart and done simultaneously.

70a - Question: Why would we think that it is forbidden?

Answer: Reish Lakish taught that Ein Ma'avirin Al Mitzvos (one may not leave one to go see the other).

Question: What Mitzvah is it to look?

Answer: "B'Rav Am Hadras Melech" (it enhances the Avodah when many people are there);

Therefore, the Mishnah teaches that there is no Isur to go to see the other.

6b: When there are no Tahor Kohanim in the Beis Av...Rav Sheshes says that Tum'as Mes is Dechuyah. We use Tehorim from another Beis Av.

33b: Dishun of the Mizbe'ach precedes Dishun ha'Menorah.

Answer (Rava): It is because Ein Ma'avirin Al Mitzvos. When one enters the Heichal, he encounters the Mizbe'ach before the Menorah.

Beraisa: The Shulchan was in the north, two and a half Amos from the wall; the Menorah was in the south, two and a half Amos from the wall; the Mizbe'ach was in the middle, slightly closer to the entrance to the Heichal than both of them.

Pesachim 64b - Mishnah: The Kohen receives a full bucket (of Dam Pesach) and then returns an empty bucket.

Inference: He does not return the empty bucket before receiving the full one. This supports Reish Lakish:

(Reish Lakish): We do not delay to fulfill a Mitzvah in front of us (returning an empty bucket is only preparation for Kabalah, it is not an actual Mitzvah).

Menachos 64b - Mishnah: L'Chatchilah, the Omer must be brought from a place close (to Yerushalayim);

If no barley near Yerushalayim is ripe, it may be brought from anywhere (in Eretz Yisrael).

Gemara - Question: Why must it be brought from a close place?

Answer #1 The Omer must be Karmel (soft. If it was brought from afar, it might dry up.)

Answer #2: Ein Ma'avirin Al Mitzvos (once we find a suitable place, we do the Mitzvah immediately.)

Mechilta (Shemos 12:17): R. Yoshiyah says, we read "U'Shmartem Es ha'Matzos" as if it said 'ha'Mitzvos'. Just like we do (delay, and) cause (dough for) Matzos to become Chametz, we do not sour Mitzvos (by delaying them). Rather, one does a Mitzvah at the first opportunity.

Rishonim

Rashi (Yoma 33a DH Ein): The Mechilta teaches that Ein Ma'avirin Al ha'Mitzvah, for then it is delayed and sours.

Tosfos (Menachos 64b DH Ibo'is) and Piskei Tosfos (Yoma 58b (36)): Ein Ma'avirin Al Mitzvos applies only when both are in front of us and there is no (other) source saying which to do first.

Rebuttal #1 (Shitah Mekubetzes 11): The Gemara says that Ein Ma'avirin Al Mitzvos applies to the Omer, even though there is only one Mitzvah to do.

Rebuttal #2 (Tosfos Megilah 6b DH Mistaber): R. Eliezer b'Rebbi Yosi says that we read the Megilah in Adar Rishon because Ein Ma'avirin Al Mitzvos, even though there is only one Mitzvah to do.

Defense (of Tosfos Menachos - Yad Binyamin Menachos 64b DH v'Lashon and DH v'Achen): Tosfos teaches that if the Torah fixed a place for a Mitzvah, e.g. on which side of the Mizbe'ach blood must be put, Ein Ma'avirin cannot determine the place. Tosfos agrees that Ein Ma'avirin applies to one Mitzvah when the Torah did not fix a place. The Malbim explains that Tosfos says that Ein Ma'avirin overrides Tadir (the more frequent Mitzvah) only when there are two Mitzvos to do. When there is only one Mitzvah, we pass over a Mitzvah to do what is more Tadir.

Yad Binyamin (ibid. DH v'Ibo'is): Tosfos asked why the Torah needed to write "Karmel". This shows that he holds that Ein Ma'avirin is mid'Oraisa. This is unlike Radvaz (1:529), who says that it is mid'Rabanan, and the Drashah in the Mechilta is an Asmachta.

Rambam (Hilchos Avodas Yom Kipur 3:8): When the Se'ir ha'Mishtale'ach reaches the Midbar, the Kohen Gadol reads the Torah and they burn the Par and Sa'ir in Beis ha'Deshen. Therefore, one who sees the Kohen Gadol read does not see the burning of the Par and Sa'ir.

Rashi (Yoma 70a DH Ka Mashma Lan) and Ri Korkus: Since seeing the Kohen Gadol read or the burning of the Par and Sa'ir is not considered engaging in a Mitzvah, going to the other is not considered passing over a Mitzvah.

Pirush ha'Mishnayos (7:2): The Mishnah (which says that there is no Isur to go to see the other) teaches that one may leave seeing one Avodah in order to see another

Note: Pirush ha'Mishnayos connotes that one may not leave seeing an Avodah to engage in Reshus. According to Rashi and Ri Korkus it is permitted. Seemingly, one who pays attention to the Kri'ah fulfills the Mitzvah of Talmud Torah! (Even though we do not bless for Hirhur, "V'Hagisa Bo Yomam va'Laylah" is fulfilled in the heart - Gra OC 47:4 DH v'Hu ha'Din.) Perhaps we discuss one who cannot hear (he is too far away) or understand the Kri'ah. Alternatively, if he would go to see the burning he would also think about Divrei Torah at the time .

Rambam (Hilchos Temidim u'Musafim 7:5): The ideal Mitzvah is to bring the Omer from nearby. If the barley nearby is not ripe, the Omer may be brought from anywhere in Eretz Yisrael.

Chidushei ha'Griz (Menachos 64b p.209 DH Sham bi'Gmara): The Gemara needed to bring both reasons for bringing from close to Yerushalayim. If the only requirement was Karmel, surely there is a Shi'ur of how far we may go without concern lest it dry up; we would not need to take the first barley encountered. If the only concern was not passing over Mitzvos, if the Sheluchim of Beis Din were far from Yerushalayim, they would take from where they are; however, since we require Karmel, they wait until they are close. It appears that it was an enactment to bring from what is nearby, hence there is no difference between the reasons.

Note: Seemingly, there still can be a difference between the reasons, even if the enactment was that the Sheluchim must start from Yerushalayim and take the first barley they encounter. Perhaps the closest barley is (far away and) prone to dry before reaching Yerushalayim, and slightly further away is barley less prone to dry. E.g., perhaps the latter grain is larger, or even though it is further away it takes less time to reach Yerushalayim because it is near a level paved road (but the path to the closer place has hills and valleys). Alternatively, if one must reap in two places because neither has enough for the Omer, it is better to reap immediately upon reaching the first place to avoid passing over a Mitzvah, but the concern for Karmel favors waiting until returning (after reaping the further place) to minimize the delay from reaping until reaching Yerushalayim.

Griz (ibid.): Perhaps both of the reasons are correct, and they do not argue with each other.

Question: The Rambam implies that it is a special Halachah to bring the Omer from nearby. Ein Ma'avirin applies to all Mitzvos!

Answer (Griz ibid.): It was an enactment to bring the Omer from nearby (even if we must pass over the Mitzvah). Alternatively, reaping the Omer is itself a Mitzvah, whereas regarding other Menachos reaping is merely Machshirei Mitzvah, and we are not concerned for passing over Machshirim.