As Ann said, Barnstormer, there are many trails in the park besides the four segments which make up the Sentiero Azzurro. Yes, at least two of the four have been closed for some time now (the one between Manarola and Riomaggiore should be fully open again in the near future), and any of them at all can close at any time due to heavy rain, rockslides or other temporary damage/safety threats. This route is also very heavily used so even erosion from all the foot traffic can require maintenance which requires occasional closure.
Other routes climb into the hills, and can be excellent hikes offering stupendous views. The park website has a complete list of trails along with current status:
http://www.parconazionale5terre.it/Esentieri-outdoor.php
Red-and-white circles indicate a closed trail. Clicking on any trail number in the list will bring up a detailed description of the route including length, average time to hike it (always add more for stopping for photos and rest stops), difficulty level, and location on the map.