Does the colony collapse? Do workers keep following the last orders? Or can future queens replace the dead one? I'm guessing this might also depend on each specific sub-species.
Not really my field, but I can point to this review which discusses a couple of different ways in which ants organize their communities: Heinze. 2008. The demise of the standard ant (Hymenoptera: Formicidae)
I know that in some species there can be several queens per colony (so more robust to queen deaths). In many species, workers can start laying eggs if the queen dies, but these will develop as males (since they are unfertilized). However, in some species (e.g. Platythyrea punctata) the workers can reproduce by parthenogenesis which can stabilize the colony to some extent after the queen dies.