1. Our parsha (16:8) tells us that matzah is eaten for six days; another pasuk tells us it is eaten for seven days. Rashi (in his first answer) explains that matzah made from chadash can be eaten for the last six days of Pesach, but not on the first day before the korban ha'omer is offered.
Gur Aryeh writes that Rashi is alluding to the chiddush din quoted in Tosfos (Kiddushin 38) that we do not say that the mitzvas aseh of matzah can be doche the lo ta'aseh of chadash. Tosfos gives two reasons why not: 1) mitzvas matzah was commanded before matan Torah and therefore has a special status; 2) if you allow a person to eat a k'zayis for the sake of the kiyum mitzvas aseh he might eat more than that, which would not be allowed. (It sounds like this second answer holds that m'dorasya matzah would be doche chadah, but m'derabbanan we do not allow a person to chance it. It also assumes that if you eat more than the shiur required to fulfill the mitzvah, there is either no kiyum mitzvah at all, or that a kiyum mitzvah without a tzivuy is not enough to be doche a lav.) The Gur Aryeh offers his own reason. The only time we say aseh doche lo ta'aseh is when it is impossible to fulfill both -- there is a direct clash between the two. For example, it is impossible for there ever to be a scenario of yibum that does not also entail a clash with the issur of eishes ach (whether yibum works through aseh doche lo ta'aseh or some other mechanism is a topic for some other time). Here, even if a person only has one matzah and it happens to be made of chadash, the Maharal does not see that as a clash. I think what he means is that the mitzvah of chadash does not logically preclude fulfilling the mitzvah of matzah. You may run into a scenarios where practically you cannot fulfill one without violating the other, but a practical roadblock is not enough.
I am a little confused by one point: is the fact that we don't say that matzah can be doche chadash derived from the fact that the pasuk says to eat matzah for six days? The rule the Maharal invokes, i.e. that you only say dechiya when there is an inevitable clash. is not unique to matzah -- it is a general rule that applies in every case of aseh doche lo ta'aseh. So why do we need a special limud here to tell us that matzah is not doche chadash?
2. The gemara (Sanhedrin 71) quotes R' Eliezer's view that if even one mezuzah is found in an ir ha'nidachas it is enough to salvage the city and spare it from destruction. (There is a view in Chazal that there can never be an ir ha'nidachat. The L. Rebbe is quoted as saying that if he heard a city was going to be declared an ir ha'nidachat, he would go there and put a mezuzah up on someone's door, so the city would never be destroyed. There is a lesson there that goes beyond the halachos of ir ha'nidachas.) The Toras Chaim and many other Achronim ask why don't we apply the rule of aseh doche lo ta'aseh? Why does fulfilling the mitzvah of destroying the city allow for the burning of the mezuzah?
One approach is that you only say aseh doche lo ta'aseh when the entire aseh is being fulfilled. Here, burning the single mezuzah is just part of the larger mitzvah, which is to destroy the entire city. (Whether this chiddush is true and why it is true is something to look into.)
Once upon a time I did a post on the Ohr Sameiach hil chameitz ch 6 who writes that you only say aseh doche lo ta'aseh on an issur gavra, not an issur cheftza. The question of Tos quoted above regarding matzah being doche chadash must assume, as the Ohr Sameiach discusses, that chadash is not an issur cheftza -- chadash is inherently heter, just cannot be eaten before the korban ha'omer is brought. The lav of "lo ta'aseun kein l'Hashem Elokeichem" which prohibits burning the mezuzah is an issur cheftza. Based R' Eliyahu Baruch Finkel suggests that therefore is not pushed aside by the chovas hagavra of destroying the ir ha'nidachas.