Q. The bracha of sheheheyanu is recited on mitzvos that come from time to time (e.g., sukkah, shofar, Yom Tov). Why is this bracha not said when making an eruiv tavshilin?

A. One may argue that you do not recite shecheyanu on a mitzva that is only a matir, or was instituted only to permit you to do something. However, Darkei Moishe (Y.D. 28: 11) mentions that one who performs shechita for the first time recites shecheyanu for covering the blood and not for the shechita itself. Yet he does not mention that the reason is because shechita is only a matir, rather because you are paining and destroying an animal.
Another reason may be that it is part and parcel of the Yom Tov itself, similar to Sefiras Haomer, where the shecheyanu on the Yom Tov includes the prior counting. The Ridvaz (4: 1327) explains that we do not recite Shecheyanu on counting Sefira since it constitutes the days of preparation and groundwork for Kabolas Hatorah on Shavuos and it is then that we do recite that brocho. (See also Birchas Hashir Vehashebach 14: 2 and 24: 3).
Horav Shlomo Miller’s Shlit”a opinion is that since there are different kinds of eiruvin, such as eiruv chatzeiror or techumim, that could be done every Shabbos and they command the same brocho of “Al Mitzvas Eiruv”, this is not considered, as a mitzva that comes only from time to time. (See Shulchan Aruch O.H. 366: 14 and Mishna Berura 79 if the bracha when doing both at the same time changes to Mitzvas Eiruvin).

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit”a