Continuing the discussion of she’hechiyanu on bedikas chameitz (link), I was also wondering about the point Anon1 raised about the mitzvah not being one of simcha and therefore exempt from she’hechiyanu. Apparently the rule of reciting she’hechiyanu over a mitzvah which occurs infrequently is a separate mechayeiv, irrespective of whether the mitzvah is one of simcha. For example, Tosfos (Brachos 37b) writes that a kohein would recite she’hechiyanu when he did avodah with his mishmar, which occurred only twice a year. I don’t know if there is a particular simcha associated with avodah; the she’hechiyanu seems based entirely on the infrequency of the mitzvah’s performance.
In terms of the disagreement between the Tur and Ba'al haIttur, I think Chaim Markowitz hit the nail on the head in distinguishing between the ma’aseh mitzvah and the mechayeiv. The Tur argued that bedikas chameitz can be performed just about any day of the year given the right conditions, e.g. a person leaving home who plans to return mid-chag and needs to make sure the house is chameitz free. Therefore, the mitzvah is not characterized as asi m’zman l’zman, something that comes at some infrequent time interval. The Ittur would argue that no matter when you do the actual ma’aseh mitzvah of bedikah, the kiyum mitzvah accomplished by its performance only takes effect on the night of erev Pesach. Or, to put it in Telzer terminology, the sibas hachiyuv is ohr l’arba asar, regardless of when the actual act of searching is done. The debate between Tur and Ittur boils down to the question of what gives rise to the need for she’hechiyanu – is it the fact that the ma’aseh mitzvah comes along rarely, or the fact that the kiyum mitzvah comes along rarely, even if the ma’aseh can be done at any time?
Anon1 raised another good point with respect to the definition of asi m’zman l’zman. Even if, as the Tur writes, the mitzvah of bedikah can theoretically be done on any day, there is still only one day in the year when it need be done. Here too, I think we get involved in chiyuv vs. kiyum. Practically the ma’aseh mitzvah is done only once annually, but the fact that there is no fixed date shows that the chiyuv is one which applies year-round. To add some fuel to the question, as I mentioned above, Tosfos holds that a kohein would recite she’hechiyanu when doing avodah twice a year with his mishmar. Yet, in theory a kohein has the option of doing avodah on any day – he could bring his own korban. That theoretical possibility of a kiyum on any given day does not preclude reciting she’hechiyanu on the twice a year days the kohein would minimally do avodah. It’s worth noting that Rashi and the Rambam offer completely different interpretations of that sugya, but even without that, I'm not sure that I'm not comparing apples and oranges -- bedikas chameitz is a chiyuv on each person; I don’t know if there is a chiyuv on every individual kohein to do avodah.
Maybe some more to come on this…