What does the Torah mean when it writes that one may not declare a firstborn animal Hekdesh?
Rashi, Rashbam and Targum Yonasan: It means that one is not permitted to take a Bechor and declare it Hekdesh in the form of another Korban (an Olah or a Shelamim), since it is not his 1 (to declare Hekdesh - Rashi).
Ramban: The Torah means (not that one may not, but) that one need not declare it Hekdesh. 2
If one cannot (Rashi), or need not (Ramban) sanctify a Bechor, why does the Torah write in Devarim (15:19) "Kol ha'Bechor ... Takdish la'Hashem Elokecha"?
Rashi #1: Although a Bechor is automatically Kadosh, it is nevertheless a Mitzvah for the owner to declare it Hekdesh 1 .
Rashi #2 (in Devarim 15:18): Although one cannot sanctify the actual Bechor as another Korban, one can however, declare its value Hekdesh and with the equivalent sum, purchase a Korban accordingly.
Ramban: What the Pasuk in Devarim means is that one should treat it as Hekdesh, not to work with it or to shear it, and to eat it in Yerushalayim.
This can perhaps be compared to the Mitzvah to add fire to the Mizbe'ach, even though fire comes down from Heaven.