Here's a summary for you, as well as a little history that everyone else seems to be forgetting. What was the issue? The Culture Ministry of Italy decided, seemingly out of the blue, to lay claim to the David which is located inside the State-run museum of the Accademia (so they already reap the financial benefits). The former mayor of Florence, Matteo Renzi, dug up documentation that proves that the state gave it to the city as a gift in 1871, along with the Palazzo Vecchio and everything found on its front porch (called a ringhiera).
So the question is: Who owns the David? The State (Italy) or the City (Florence). And who cares? Well, the city cares, because the state received 8 million euros in ticket sales thanks to the David, but they told the City that it is responsible for cleaning the wall of the museum that has been covered by the graffiti of waiting tourists. If the statue belongs to the city, it'll clean the wall, but will also want a cut of ticket revenues. If the State is so convinced of its ownership, it ought to also clean the wall.
But there is one more player here that everyone forgets. The NYT states that "the Florentine Republic [...] commissioned the statue in 1501". This is imprecise. The commission was by the Operai del Duomo, or more specifically the Overseers of the Cathedral works. A document of July 2, 1501 records these Overseers determined to find a sculptor to finish the David that was begun in 1464 as part of a program to decorate the Cathedral facade wtih statues of Prophets. On August 16, 1501, we read that Michelangelo has been commissioned to do the job. He worked on the marble in the workshops of the Opera del Duomo which are still visitable today.
The David was made to be placed on a buttress high up on the front of the Duomo. Only after it was completed and determined to be too good to "hide" up there was the decision made to put it in front of Palazzo della Signoria (now Palazzo Vecchio), the seat of the Republican government of Florence. This decision was made in a much-cited debate of January 25, 1504 (modern dating), which is recounted in the journal of Luca Landicci, a citizen of Florence.
A meeting was held at the office of the Opera del Duomo (a location that confirms their prime role in the matter), to which were called many artists (including Andrea della Robbia, Piero di Cosimo, Leonardo da Vinci and Lorenzo di Credi). Landucci says he's writing down their opinions "word for word" as he tells us who suggested which location. In the end, the gentlemen agreed upon the location in front of the government's palazzo and the Opera hired a guy to transport it, which we learn took four days and forty men using a "complicated mechanism of ropes".
As far as we know, there is no document that signs over the work from the Opera del Duomo - who paid for it to be made - to the city of Florence. If ownership is related to commission and payment, one could argue that the Opera del Duomo owns the David.
The Opera del Duomo has not laid claim to the David, nor is it likely too, but this proposal seemed to be just as likely (and ridiculous) as the argument between the State and the City. It should be said that the Opera del Duomo at times took care of large building projects on behalf of the city and funded by the citizens, so the city and the Opera could be considered to be in partnership and, in some ways, one and the same legal entity.
Hibbard, Michelangelo (2nd ed., p. 52f) Klein and Zerner, Italian Art 1500-1600, Sources and Documents (p. 39-44) Parks, 'The Placement of Michelangelo's David: A Review of the Documents', Art Bulletin, LXVII , pp.560-70