I will be staying in Corniglia this June 2020. Are all the hiking trails going to be closed indefinitely?
As I understand it, the high trail between Corniglia and Manarola is open (via shuttle bus to Volastra). And the popular lower trails that connect Corniglia to Vernazza to Monterosso is open, as well as the trail from Levanto to Monterosso too. Of course, all this is subject to change between now and this summer.
I’ll be traveling there in June as well, and will get the official scoop when we arrive.
They're not all closed right now. Only the trails marked in black on the park's map are currently closed, and I would expect 2 currently closed legs of the popular SVA2 "Blue Trail" to be open by spring. Those would be the sections between Monterosso and Vernazza, and between Vernazza and Corniglia. The "Blue Trail" sections from Corniglia to Manarola and Manarola to Riomaggiore will not be open.
http://www.parconazionale5terre.it/Esentieri-outdoor.php
Understand that trails can close at any time due to heavy rain, rockslides and general damage so it's impossible to say with 100% certainty what will be open on any future day.
As Kathy mentioned, the two northern segments of the Sentiero Azzurro trails should be open at that time, although that could change due to storm damage or whatever. You will need a Park Pass to hike those particular trails (easily obtained when you get there).
As the map shows, there's an extensive network of trails in that area but many of the others go higher in the hills so they're a more robust hike.