The Biblical mitzvah of lulav applies outside the Mikdash for only the first day of Yom Tov, but as a zecher l’mikdash, in remembrance of the Mikdash which we no longer have, we take lulav for all 7 days of the holiday. The gemara (Sukkah 46) quotes R’ Yehoshua ben Levi that only the first day is “mitzvas lulav”, but the remainder of the days are “mitzvas zekeinim”. Rashi explains that RYbL held that the bracha on lulav should be recited only on the first day of Yom Tov, but the bracha of netilas lulav should not be recited on the days instituted as a zecher l’mikdash. Rav disagrees, and holds that the bracha of lulav should be recited on all 7 days.
Perhaps the gemara’s debate can be explained in light of the Brisker Rav’s (Menachos 66 stencil) suggestion of two different possible understandings of the concept of zecher l’mikdash: 1) despite the loss of the Mikdash which would nullify certain mitzvos, the Rabbis instituted that the nullification be ignored and the same mitzvah continue for posterity as a remembrance; 2) the destruction of the Mikdash did nullify the original mitzvah, but the Rabbis instituted new acts that were similar to the original nullified mitzvah to serve as a remembrance of what was.
If one adopts the first approach, then the same mitzvah of lulav which was in force for 7 days at the time of the Mikdash continued even post-destruction and the same bracha of netilas lulav should be recited. However, if one holds that that the mitzvah of lulav was cancelled with the destruction of Mikdash and all we have left is a new mitzvah to remember what once was, then this new mitzvah of zecher l’mikdash should not require a bracha of netilas lulav, as the mitzvah being fulfilled is not one of netilas lulav, but simply the mitzvah of making a zecher l’mikdash.