BACKGROUND: Recently during Sukkot, and the time to shake the Lulav and Etrog a non-Jewish "member"of the Congregation felt insulted, because he did not have his own, and our Rabbi would not allow him to borrow one to use. Obviously depending on the answer to my other question about a non-Jew being required to do the Mitzvot; this may nat require any further consideration.
Your Rabbi was most likely following the halacha as explained by Rambam in Mishnah Torah, Sefer Kinyan, Hilchot Zachiyah v'Matanah, Chapter 3:11 which prohibits a Jew from giving a gift to a non-Jew.
אָסוּר לְיִשְׂרָאֵל לִתֵּן לְעַכּוּ''ם מַתְּנַת חִנָּם. אֲבָל נוֹתֵן הוּא לְגֵר תּוֹשָׁב. שֶׁנֶּאֱמַר (דברים יד-כא) "לַגֵּר אֲשֶׁר בִּשְׁעָרֶיךָ תִּתְּנֶנָּה וַאֲכָלָהּ אוֹ מָכֹר לְנָכְרִי", בִּמְכִירָה וְלֹא בְּמַתָּנָה. אֲבָל לְגֵר תּוֹשָׁב בֵּין בִּמְכִירָה בֵּין בִּנְתִינָה מִפְּנֵי שֶׁאַתָּה מְצֻוֶּה לְהַחְיוֹתוֹ. שֶׁנֶּאֱמַר (ויקרא כה-לה) "גֵּר וְתוֹשָׁב וָחַי עִמָּךְ":
"It is forbidden for a Jew to give a gift to an Akum (Literally one who is under the dominion of the stars and constellations. This means a gentile who has not accepted the 7 commandments of Noach.). But one can give to a Resident Alien (meaning one who accepted the 7 commandments of Noach and who resides in the land of Israel). As it says (Devarim 14:21) To the stranger who is in your gates, you will give it and he will eat it, or sell to a gentile.; through a sale and not through a gift. But to a Resident Alien whether through a sale or through a gift: Because you are commanded to sustain him. As it says (VaYikra 25:35) (who is) a stranger and a resident and lives with you."
עכו"ם שאינו גר תושב אסור ליתן לו מתנה אא"כ הוא מכירו או אם יש בו בדבר משום דרכי שלום
מתנה. אם לא שיש לו ג''כ צד הנאה מהעובד כוכבים דה''ל כמכירה. סמ''ע:
"It is forbidden to give an Akum, meaning that he is not a Ger Toshav (a Resident Alien), a gift. Rather, we say thus, it is a sale to him (to the gentile) or if the transaction has some beneficial aspect (for the Jew) in the transaction.
Ba'er Heitev: It is a gift if there is not also some aspect of benefit to the Jew from the Akum, that thus it is like selling. (As found in Sefer Meirat Einayim)"
When one Jew gives his lulav to another Jew in order to fulfill the mitzvah, it is given as a gift, meaning it becomes the full property of the other Jew. The concept of giving a gift in order that it be returned is only between one Jew and another.
That special type of gifting, which means that the act of the giving is only accomplished when the prerequisite is accomplished, is only between Jews. In the context of the lulav, that means that by fulfilling the requirement of returning the lulav, it becomes retroactively the property of the Jew who received the gift. And only then does the one who received the gift receive the benefit of the actual fulfillment of the mitzvah.