Family search has the (in Italian) marriage banns, marriage records and death indices from 1866 forward for Neirone under the regional administration office in Chiavari that you can browse for free. Other other comune/Frazione in the region have some info available also.
https://familysearch.org/pal:/MM9.3.1/TH-1942-21543-23113-79...Ancestry has more (browseable)records, including birth, but you will need to pay to access the "Italy" part of the databases.
http://search.ancestry.com/search/db.aspx?dbid=1895Unfortunately, neither have the church records which would take you further back.
Good luck,
Chris Palmer