Reverend Mike Barnsley of St John's Church in Cinderford, Gloucestershire, said damage caused to his graveyard by the animals has caused "upset and distress".
Some residents claim people are feeding the boar which encourages the animals into towns and villages.
West Dean Parish Council said it has formed an action committee.
Tim Gwilliam, district and parish councillor for Berry Hill, said culls are not working.
He said: "Perhaps it's the tactics that need addressing. We need to put it right.
"There is nothing worse than somebody's grave being turned over. It's horrendous. It's 2017 and we can't allow this to happen."
Rev Barnsley said it is "very upsetting" for families who have visited the graves of their loved ones at St John's to see "this sort of mess".
Despite years of culling, the last count put the number of boar in the region at nearly 1,600 - more than 500 up on the year before.
The target population for the Forestry Commission is 400.
Wild boars were hunted to extinction in England in the 17th century, but 12 years ago about 60 farmed animals were illegally dumped in the Forest of Dean.
The Forestry Commission said six wildlife rangers have been deployed on this season's cull, and details will be released when it finishes later in the spring.
A council has formed a committee to tackle wild boar digging up a graveyard.