Sometimes, when a machine gun turret is destroyed, it will not respawn. However, the turret's stand will remain operable. Players can fire such destroyed turrets; the bullets will appear to come out of thin air. However, the turret itself (which is not there) will not shield the operator from enemy fire.
On a side note, in Halo 3, you can destroy a Scorpion's coaxial machine gun, and it will still be fully operable. In addition, on the Halo 3 level The Ark, there is a destroyed LAAG-equipped Warthog with an operable, yet destroyed, machine gun turret.