After the Akeida on Mt. Moriah, Avraham named that site "Hashem Yir-eh" which means Hashem sees all. Then he continues, "On this mountain Hashem will be seen" (B'reshis 22:14).
According to the Rambam Hilchot Beit HaB'chira 7, even today we are prohibited from walking on Mt. Moriah, where the Mikdash stood. The reason is that this place still retains its Kedusha even in the absence of the Mikdash. No power in the world can neutralize the holiness of this spot.
Rav Soloveitchik asks in Derashot Harav why is there no similar prohibition to ascend Mt. Sinai, where the Torah was given? Mt. Sinai maintains no Kedusha following the awesome event of the giving of the Torah. Why is there this difference in Halacha between these two mountains?
Rav Soloveitchik answers that we did not actively participate in the giving of the Torah. According to the Midrash Shir HaShirim Rabba, we overslept on the night of Shavuot and Moshe had to wake us up in order to receive the Torah. We did nothing to contribute to this blockbuster event. Thus, despite Mt. Sinai having been the site of the greatest Divine Revelation in human history, none of its Kedusha remains. As soon as the Divine Revelation was over, the holiness of the mountain vanished as well.
Mt. Moriah, explains the Rav, was completely and totally different. Avraham prepared an altar on that mountain and was willing to sacrifice his son on it. Thus, Avraham became a partner with Hashem instilling permanent Kedusha on Mt. Moriah, which is the Temple Mount. As the verse states in B'reishis 22, "This place is the eternal resting place of Hashem's Presence."
This explains why there is no Chabad House on Mt. Sinai. And we are not even certain where Mt. Sinai is because we did not contribute to its Kedusha. The message of the Akeida is that we must be Hashem's permanent and active partner in Tikun Olam.