Does anyone have current information on how much of the trail is closed?
Jeff,
At the present time, the only Sentiero Azzurro trails that are open are the Monterosso-Vernazza and Vernazza-Corniglia segments. The other two segments (Corniglia-Manarola and Manarola-Riomaggiore) are not expected to re-open for at least a year (and perhaps longer). There's an extensive network of trails in that area and it's possible to bypass the closed segments using other trails that go higher into the hills (or you can use the local trains). Keep in mind that the status of the trails can change at any time, due to storm damage or whatever.
Note that a Park Pass is required to hike the Sentiero Azzurro trails. You can obtain that easily at the small wooden huts at the beginning of the trails.
Jeff, the park has a good website that you can check for trail info and status:
http://www.parconazionale5terre.it/Eindex.php
This is the page with the map and individual descriptions of each trail: click on any segment on the map of the listings below it to find out length, difficulty rating, etc. A red-and-white circle means a trail is closed:
http://www.parconazionale5terre.it/Esentieri-outdoor.php
The "Blue" trail encompasses the 4 segments from 592-1 (SVA2) Riomaggiore - Manarola to 592-4 (SVA2) Vernazza - Monterosso.