Like many other sanctuaries throughout the ancient world, the temple of Melqart probably constituted a place of asylum---a status that Alexander seems to have respected on this occasion. Of course, he may have also desired to uphold the aura of his original pretext for entering Tyre, i.e. to peacefully make sacrifices to the god.

I owe the asylum explanation to an article by Jonas C. Greenfield ("Asylum at Aleppo: A Note on Sfire III, 4-7," in Ah, Assyria... Studies in Assyrian History and Ancient Near Eastern Historiography Presented to Hayim Tadmor, Scripta Hierosolymitana 33, ed. Mordechai Cogan and Israel Eph‘al [Jerusalem: Magnes Press, 1991], 272-278, esp. 276). As a bonus, I can also recommend an excellent article on the religious context surrounding the siege: Ory Amitay, "Why did Alexander the Great Besiege Tyre?" Athenaeum 96 (2008): 91-102. :)