So for example, when the gemara suggests that you should be able to violate the lav of breaking the bones of the korban pesach in order to get to the meat inside and be mekayeim the aseh of achilas korban pesach, it makes perfect sense -- the lav of breaking the bones is incidental to eating the meat.
Another example of the rule in action: The Rambam writes that one cannot eat matzah made of tevel:
אין אדם יוצא ידי חובתו באכילת מצה שהיא אסורה לו, כגון שאכלה טבל, או מעשר ראשון שלא ניטלה תרומתו, או שגזלה. זה הכלל--כל שמברכין עליו ברכת המזון, יוצא בה ידי חובתו; וכל שאין מברכין עליו ברכת המזון, אין יוצא בה ידי חובתו.
Putting aside the strange formulation the Rambam uses -- connecting this din to the issur of bracha on ma'achalos assuros -- the Sha'agas Aryeh asks why we don't say aseh doche lo ta'aseh in this case. Why does the mitzvah of eating matzah not push aside the lav of tevel?
The Ohr Sameiach answers that tevel is an issur achila that tells you not to eat the very same foodstuff that the mitzvah of matzah tells you that you should eat. In this case there is a direct clash betweeh the aseh and the lav, and therefore aseh doche lo ta'aseh doesn't apply.
The obvious question is that the two most famous cases in shas of aseh doche lo ta'aseh don't seem to fit the rule: 1) Yibum -- the aseh is to be meyabeim the very same woman which the lav prohibits. 2) Kilayim b'tzitzis -- the very same garment that you put on to be mekayeim tzitzis is assur to wear because of the lav of kilayim. How do these cases work?
The O.S. explains that we need to distinguish between two different types of mitzvos aseh. The miztvah of matzah is on the "karkafta d'gavra;" the person has to find a piece of matzah to eat, but there is no rhyme or reason for it has to be piece A as opposed to piece B. Instead of choosing a piece that is tevel, the person can just as easily take a piece that is non-tevel.
The mitzvah of yibum on the other hand does not require that a person go out and find a woman to be meyabeim; it demands that a specific woman in a specific circumstance have the mitzvah of yibum be fulfilled with her. Similarly, there is no mitzvah to run out and find a four cornered garment to wear; the mitzvah is to put tzitzis on a four cornered garment if you happen to choose to wear one.
Where the Torah demands that the mitzvah be done with a specific item, then you can use aseh doche lo ta'aseh to accomplish the goal -- the aseh that relates to the specific item overrides the lav inherent in the same item. However, when the mitzvah is on the "karkafta d'gavra," then you can't invoke aseh doche lo ta'aseh -- instead, go get a different piece of matzah.
I was telling someone this O.S. to my son in yeshiva and someone standing nearby asked asked what the question as to why the mitzvah of hashavas aveidah does not allow a kohein to enter a cemetery to return a lost object because of aseh doche l"t. I answered that this case fits the pattern perfectly. There is no mitzvah on the "karkafta d'gavra" to go looking for an aveidah to return -- the mitzvah is to see that an object once found is returned to its owner. This case is just like yibum and kilayim.
Tosfos in a few places in shas asks why a chatzi eved cannot marry a shifcha based on the aseh of p'ru u'revu being doche the lav of k'deisha. The answer, writes the O.S., is simple: the mitzvah of p'ru u'revu is on the "karkafta d'gavra." Who says the chatzi eved needs to marry this particular woman?
Final point: Tosfos (Kiddushim 39) asks why the aseh of achilas matzah is not doche the lav of eating chadash. The Yerushalmi (quoted by Tosfos) answers that the aseh of matzah is weaker than the lav because it was given before mattan Torah. Based on the O.S. the question should not even get off the ground -- how is this case any different than the case of matzvah made of tevel? Why not just answer that the aseh that is on the gavra does not doche a lav inherent in the cheftza?
The O.S. answers that the difference between tevel and chadash is that while tevel is inherently assur, like ma'achalos asuros, chadash is assur only because the korban ha'omer has not yet been brought -- it's not a cheftza d'isura, bur rather simply food that is waiting for it's matir to arrive (see the O.S. who proves that this is actually a machlokes Bavli and Yerushalmi.)
Maybe more on this next week -- it's a brilliant piece on a shas sugya worth taking the time to see inside over Shabbos.