Is the Halo a persistent universe?

Not strictly speaking, at least not in the sense that objects are persistent in massively multiplayer online games like Ultima Online or their ilk, although it was speculated that Halo might utilize such an architecture early in its development.

Bungie has stated that within the solo campaign, objects are persistent in the sense that if the player sacks and destroys a Covenant base, it will stay that way unless there's action taken to change that.

In short, we can expect that if we leave an area in Halo and then return, things won't magically be fixed if we've destroyed them, and Covenant aliens won't be respawning randomly all over the place.