Do these places possess an intrinsic, metaphysical sanctity, embedded in them since Creation? That is the mystical view presented by R. Yehudah HaLevi in the Kuzari (V:23). It is the view taken as a given by countless rabbinic authorities over the ages, and popularly assumed today to be the only conceivable approach.
Rambam, on the other hand, was of the view that the sanctity of these places is not a metaphysical quality. Rather, it is a status that stems from their historical role. When Rambam stresses that the site of the altar must never be moved, the reason that he gives is not that it possesses inherent metaphysical significance. Rather, it is because of the history of the site, in terms of the events that took place there—the placement of the altar there by David and Solomon, the usage of that site by Avraham, and so on (Mishneh Torah, Hilchot Beit HaBechirah 2:1-2).
And the original selection of these sites could have been for relatively mundane reasons. Rambam's explanation for the selection of the Temple Mount will no doubt come as a shock to many:
"It is known that idolaters sought to build their temples and to set up their idols in the highest places they could find there, on the highest mountains. Therefore Avraham Avinu selected Mount Moriah, because of its being the highest mountain there, and proclaimed on it the unity of God." (Guide for the Perplexed 3:45)
Within Mount Moriah, Avraham decided that any divine worship would take place facing the west, and the the Temple itself was eventually situated there. The reason for this was again not connected to any special metaphysical properties of the westernmost part, but for a different reason entirely:
"Avraham designated the western part of it, that the Holy of Holies would be in the west… And it appears to me that the reason for this was that the popular view in the world at that time was to worship the sun as a god, and so people undoubtedly turned in prayer to the east. Therefore, Avraham Avinu turned to the west on Har HaMoriah—that is to say, in the Sanctuary—in order to have his back to the sun." (ibid.)
The consequence of Rambam's view, that the sanctity of the Land is a function of its usage rather than due to any intrinsic metaphysical qualities, is that this sanctity can disappear:
"All territories held by those who came up from Egypt, and consecrated with the first consecration, subsequently lost their sanctity when the people were exiled from there, since it was consecrated at the time due to the conquest alone and was not consecrated for all time. When the exiles returned and seized part of the land, they consecrated it a second time with a permanent consecration, both for that time and the future. " (Mishneh Torah, Hilchot Terumot 1:5)
This view on the nature of the sanctity of the Land of Israel is not unique to Rambam, nor to the medieval philosophers. The same view is to be found in the writings of Rav Soloveitchik:
"With all my respect to the [views of certain] Rishonim, I must disagree that kedusha is an objective metaphysical quality inherent in the land. Kedusha… is man-made; more accurately, it is a historical category. Soil is sanctified by historical deeds performed by a sacred people, never by a primordial superiority. Kedushat Ha’aretz denotes the consequence of a human act." (Rabbi Joseph B. Soloveitchik, The Emergence of Ethical Man, p. 150)
It is crucial to stress that this does not mean that according to the rationalist approach, the Land of Israel or Jerusalem or the Temple Mount are any less holy than according to the mystical approach. Rather, it is simply a different perspective on what the nature of holiness is all about.
(Adapted from my forthcoming book Rationalism vs. Mysticism: Schisms in Traditional Rabbinic Thought. For extensive discussion, see Menachem Kellner's important work, Maimonides' Confrontation with Mysticism, pp. 107-115. He also references numerous other studies on this topic.)