First of all currently there are NO RESTRICTIONS TO VISITORS' ACCESS to the towns.
I'm saying this because every now and then there are posters that continually fish out the same stale article that came out in February 2016 on the statements made by the President of the Cinque Terre Park (Mr. Vittorio Alessandro). Zombie posts, like zombies, are hard to kill.
The latest is that Trenitalia has instituted shuttle trains, called 'Treni Cadenzati' (staggered trains) aka the "Cinque Terre Express" every 30 min in both directions between La Spezia and Levanto. This service started last December.
The cost is 4 euro per person if boarded between La Spezia and Levanto. Those who board at farther stations, will pay an additional 2.40 euro over the price of the regular ticket. Residents with ID attesting their residence in the area, are exempt from these fares, which are higher than normal.
Since the end of April, Trenitalia has staffed the stations with 'body guards' (lets call them that way), which will limit the number of passengers who can access the platforms on the most critical days (presumably weekends). So only a limited number of people will be given access at once on the platform, de facto limiting the number of people who can board a regional train at once. That might result in lines outside the station before you can get on a train.
There are no other restrictions or limits in place, they are just trying to move the long queues from the platforms (near the tracks) to the outside of the station, for safety reasons.
The Via dell'Amore is still closed. The Via dell'Amore is not property of the CT National Park, but of the City of Riomaggiore. The City is moving to accept bids from companies willing to rebuild the path (est. cost=16 million euro) in exchange for the exclusive right to manage the path and collect the admission fees. Last month the President of the Park (Vittorio Alessandro) has criticized the move. The Mayor of Riomaggiore has responded that the President of the Park "should mind his own business" since he has no jurisdiction over the Via Dell'Amore.
This Mr. Alessandro, being a politician, obviously likes publicity and every month he is in the news giving these interviews, which are regularly criticized by local politicians. The Governor of Liguria (who belongs to the opposing political party) dispelled Mr. Alessandro's proposal of limiting access to the Cinque Terre as hogwash last February.