It says in the Torah, that a tereifah must be thrown to the dog.
Why is this not counted as one of the 613 mitzvos?
This is a good question. From the sources below, we will see that although the Torah writes it as an obligation, the instruction was interpreted not as a command, but as permission to give the meat to the dogs.
The Minchas Chinuch (73:3) asks this question: Surely the Torah writes that a tereifah must be thrown to the dogs. Why, if so, do the poskim in general, and those who list the mitzvos in particular, make no mention of this mitzvah?
Indeed, the Rema (based on the Tur), Yoreh De’ah 117, writes that one can sell a tereifah to a non-Jew, clearly implying that there is no obligtation to give the tereifah to a dog. The Chikrei Lev (Yoreh De’ah 19) expounds on this matter at length, and explains that there is no obligation to give the non-kosher meat to a dog.
The Darchei Teshuvah (29:2) explains that according to the poskim who make no mention of the obligation, the verse does not meat to imply an obligation to give the meat to a dog, but only to permit the meat to be fed to dogs. This interpretation is also given the Ibn Ezra (Yesod Morah, Sha’ar 2), as cited by Rav Yerucham Perelow (intro. to Mitzvos of Rasag, no. 6).
The Gemara (Pesachim 22a; Kiddushin 58a) used the verse “you shall throw it to the dogs to extract a halachah for kodshim: Only a tereifah can be thrown to the dogs, but chullin that were disqualified by being slaughtered in the Sanctuary cannot be thrown to the dogs.
Although Rashi (Pesachim) writes that without the word “Oso” the pasuk means to tell us the mitzvah of giving the tereifah to the dogs, which is their reward for not barking in Egypt, the Sefas Emes explains that this should not be undestood to mean that there is actually a mitzvah to feed the meat to the dogs.
It is noteworthy, however, that Tosafos (Yomah 36) imply that there is a concrete mitzvah to feed the tereifah to a dog, and this is mentioned by the Minchas Chinuch.
Another source for the mitzvah is the teaching of the Yerushalmi. Speaking of somebody who used to feed tereifos to Jews, and when he fell off the roof to his death, dogs gathered round and licked his blood, the Yerushalmi (Terumos 8:3) writes that the person “stole from the dogs, and fed Jews.” However, this is not a strictly halachic source, and it does not constitute a full proof.
Assuming, as all poskim do, that there is no actual mitzvah of feeding tereifah meat to dogs, we find an interesting example of how the Sages can treat similar-sounding verses very differently: One becomes a Torah obligation, whereas another is merely “permission” to do something.