The League Two side finished 90th out of 92 in the Football League and did not win any of their last 11 games.
Feeney has offered terms to eight players and taken up one-year options on 19-year-olds Tom Owen-Evans and Kieran Parselle.
Mark Byrne, Andrew Hughes, Alex Rodman, Danny Holmes, Darren Jones, Scott Boden, John-­Christophe Ayina and Medy Elito have all been offered new terms.
However, Yan Klukowski, Matt Partridge, Seth Nana-­Twumasi, Rhys Taylor, Joe Green, Nathan Ralph and Scott Barrow have been released.
Loanees Janoi Donacien, Ben Davies, Souleymane Coulibaly, Dean Morgan, Tommy O'Sullivan and Deanne Smalley return to their parent clubs.
First team strikers Lenell John-Lewis and Tom Meechan and goalkeeper Joe Day are already contracted for next season.
Academy graduates Lewis Bamford, Dafydd Jones and Ben Jones - brother of defender Darren - and Wales Under-19 international Liam Angel have also agreed terms to remain at Rodney Parade.
Newport County boss Warren Feeney has allowed 13 players to leave the club.