The Talmud rules that one who moves from a house may not remove its mezuzot if a Jewish person will be moving in.[1]
There are several ways of explaining this prohibition. Some Rishonim explain that once a mezuzah is placed on a Jewish door, the sanctity of the Divine Presence is resting on the room. Removing it unnecessarily would be inappropriate.[2]
Others write that since a mezuzah has the power to guard against the intrusion of negative spiritual energies (mazikin), one who removes it is potentially damaging the one who will be moving in and may not put up mezuzot on time. The Talmud records that by callously disregarding the safety and welfare of others, one could be exposing his own family to potential danger.[3] Of course, if the new owners of the house are not Jewish, the mezuzah should be removed.[4]
Incidentally, sometimes after remodeling or after a move, one may find oneself with an extra mezuzot. It is praiseworthy to keep them actively in service by loaning them out or selling them.[5] Some communities have mezuzah gemach services that loan out mezuzot temporarily.
[1] See above in the introduction “A Deeper Look”, section “Mazikin”.
[2] Ritva, cited in Nimukei Yosef, Bava Metzia 102a; Agur B’ohalecha writes that the concern is that perhaps the new dweller will not put up the mezuzot on time.
[3] Tosafos, Bava Metzia 101b; Shitah Mekubetzes, Menachos 41b note 24 cited in Agur B’ohalecha 40:1.
[4] Shulchan Aruch Y.D. 291:2.
[5] Agur B’ohalecha 40:6.