Pretty much what it says in the tin. Can the DFA learn Warlock invocations and vice versa?
Or are they separated by class? I know there are some overlaps in invocations.
Invocations are class specific - in particular, they're not the same type of invocation. Dragonfire Adept invocations are "draconic invocations." From the class description:
A dragonfire adept has a repertoire of attacks, defenses, and other abilities known as draconic invocations, which allow her to focus the draconic energy that suffuses her soul. A dragonfire adept can use any invocation she knows at will.
The section on invocations later in the book then describes the ones intended for Dragonfire Adepts as "draconic invocations."
Similarly, the Warlock's class description (Complete Arcane, p. 5) says:
A list of available invocations can be found following this class description, and a complete description of each invocation can be found in Chapter 4 of this book
...implying that invocations found elsewhere in the rules aren't available to Warlocks unless they specifically say they are.
However, all is not lost! Wizards published the Infernal Adept feat in an online article, which allows Dragonfire Adepts to learn Warlock invocations and vice versa:
Prerequisites: Ability to use greater invocations as a dragonfire adept or a warlock, special
Benefits: You learn one invocation from another class's list, choosing an invocation no more than two grades lower than the highest grade of invocation you know. For instance, an 11th-level warlock could learn a least invocation from the dragonfire adept list, while a 16th level dragonfire adept could learn a least or lesser invocation from the warlock list.