The Yerushalmi (Shabbos ch 14) asks why it is that if someone tears kriya on Shabbos he is yotzei, despite having been mechalel Shabbos, but if someone steals matzah and eats it he is not yotzei. The Yershallmi answers, "taman gufa aveira." R' Chaim interprets the Yerushalmi to mean that mitzvah haba'ah b'aveira is a psul in the cheftza shel mitzvah. There is no cheftza shel mitzvah when it comes to kriya -- there is a chiyuv on the person to tear whatever he happens to be wearing, not a din on the clothers to be torn -- but there is when it comes to matzah.
The Rambam adopts this view of the Yerushalmi. In Hil Shofar the Rambam paskens that one is yotzei tekiyas shofar using a stolen shofar because "ain b'kol din gezel." Even though an act of theft was involved in the kiyum hamitzvah, one is still yotzei because the cheftza shel mitzvah is not the shofar -- it's the kol, the voice, and that is not something that can be stolen.
This is a good opportunity for me to make up for my not having shared a beautiful pshat in a Rambam that I heard from R' Naftoli Jeger, R"Y of Shor Yoshuv, in his Hoshana Rabbah shiur this year. In brief: the Rambam holds that a stolen lulav is pasul only on the first day of Yom Tov but not other days (other Rishonim disagree). The difference between the first day and all others seems to be based on the fact that netilas lulav on the first day is a din d'oraysa, "u'lekachtem lachem bayom harishon," while only derabbanan on other days. However, the Rambam himself holds that aside from the mitzvah of "u'lekachtem lachem" there is a mitzvah d'oraysa of "u'smachtem lifnei Hashem Elokeichem," to take lulav and esrog in Yerushalayim for all seven days. Why then does the Rambam not mention that there is a psul of lulav hagazul for all seven days in Yerushalayim? R' Jeger quoted (I forget exactly who) that the answer may be that even though "u'smachtem lifnei Hashem" is fulfilled using a lulav, the cheftza shel mitzvah is actually the simcha which is engendered by that act (similar to shofar, where the cheftza shel mitzvah is the kol, not the shofar used to produce it.)
Last week we discussed (link) the Ohr Sameiach's answer to why we don't say aseh doche lo ta'aseh when it comes to eating matzah made of tevel and instead classify it as a mitzvah haba'ah b'aveira. Based on this R' Chaim, it is possible to say a different answer. The Rishonim (end of Yoma) debate the following case: a person is deathly ill on Shabbos and requires meat. Non-kosher meat is readily available. Is it better to feed the sick person that non-kosher meat, or to slaughter an animal on Shabbos in order to obtain fresh kosher meat? At first glance it would seem to be preferable to feed the person non-kosher meat, as neveilah is only an issur lav, which is less stringent than Shabbos. However, one can make the argument that once there is an allowance to violate Shabbos for pikuach nefesh, the food prepared in 100% kosher with no taint of issur. Not so with neveilah, which is inherently ma'achalos asuros and remains so. The fact that you have an allowance to eat non-kosher food under the circumstances doesn't magically transform neveilah into a a properly shechted roast beef. (See previous post here.)
Similarly, the Rambam holds that even if one is allowed to eat non-kosher food, one would not recite a bracha over it. The special circumstance that allows the food to be eaten does not transform it into a different physical substance -- it's still ma'achalos assuros, albeit with a heter.
Aseh doche lo ta'aseh works like the heter to violate Shabbos to save a life -- it lifts the lav that blocks the goal of fulfilling the mitzvas aseh. However, it cannot transform a cheftza shel issur into something different. We now understand why matzah shel tevel remains prohibited. Since it is a mitzvah haba'ah b'aveira, which the Rambam defines as a cheftza of issur (a point the Rambam reinforces by the comparison to hilchos brachos), the rule of aseh doche lo ta'aseh is useless, as the cheftza of issur remains fundamentally unchanged and unusable for the purpose of a kiyum mitzvah.
(In his Reshimos Shiurim on Sukkah R' Reichman discusses this R' Chaim at length, but suggests a different approach to this Rambam that I don't quite understand. I am not sure why he did not understand it the way I presented it here, which is based on R' Baruch Mordechai Ezrachi's shiurim. How to work this approach out so that the kilayim b'tzitzis case fits is a story for another post.)