Considering that the Lannisters currently control the Iron Throne, how does the crown’s debts to Casterly Rock work? Are the Lannisters now in debt to themselves?

As far as House Lannister is concerned, the Iron Throne belongs to House Baratheon; indeed, they must uphold that fiction at every point if they are to have any pretense of legitimacy. Hence, the debts of Robert Baratheon are passed on to his heirs, Joffrey and Tommen.

Now, it is entirely possible for Tywin to forgive those debts, as Tyrion asks him to. Tywin’s refusal is more personal in nature; he’s spent his entire life paying the king’s debts, first with Aerys II and then with Robert, and he views the reign of King Joffrey as where he will be repaid in every sense of the word. 

Tywin’s refusal, and Cersei’s blind spot in this matter, points to a certain failure of imagination on the part of the Lannisters, that they don’t think of the Iron Throne and House Lannister as part of a single institution – in which case, the best strategy would be to forgive the Iron Throne’s debt, massively improving the Crown’s financial standing and allowing them to deal with Braavos and the Faith without need for default or concession – but rather see the Iron Throne as a means of extracting power and wealth for House Lannister.