I've been following the saga of the new airport (same location but new 8000 ft long runway and new terminal) and it appears that all various State, Regional and Civil Aviation authorities have approved the master plan. The runway is still not long enough for transoceanic flights on wide-bodies, but at least the intra-European flights will be on A320 and B737 and won't be subject to so many weather related cancellations.
There are still a few steps before work on the new runway can commence. That includes fighting environmentalists in court and beating the opposition of some nearby towns (fearful of the increased noise). The opposition of Pisa airport (which doesn't like the competition) seems to have subsided since the operators of both airports are now under the same ownership. The Italian Government's goal is to have it open for 2017, when Florence supposedly will host the G-8 summit.
For those of you who can read Italian, here are the details, otherwise you can look at the pictures.
http://www.associazione-aeroporto-firenze.it/wp/wp-content/uploads/2014/11/Aeroporto69.pdf