Accident happened in a sport field on Saturday morning where a church was handing out food to the poor, some of whom had waited there overnight.
The crowd “became tumultuous and uncontrollable” and organisers failed to calm the situation, police say.
Many of those who died in the crush were women and children.
Witnesses told the AFP news agency there was frantic pushing and some were trampled underfoot when people trying to reach the entrance were forced back.
“They were telling people ‘Go back, go back, go back,'” Chisom Nwachukwu said. “Some people that were pushing from backwards were marching on those people.”
Security and emergency services were called to the scene to try to bring the situation under control. Some of the injured are being treated at Port Harcourt’s military hospital.
The King’s Assembly Church said it was “deeply saddened” by the incident, and was reviewing its safety and crowd management procedures.
It started the annual charity in 2014 but said this year’s “turnout, build up and the attendant circumstances were absolutely unanticipated”, and that attendees started gathering before security teams for the event were on site.
Rivers state police commissioner, Eboka Friday, has called on religious and charity groups to “ensure they work with the police for security and crowd control” in organising such programmes in the future.
In addition to the police investigation, a regional government committee will be formed to look into the circumstances of Saturday’s incident, says Rivers state governor Nyesom Wike.
Nigeria has seen several similar tragedies in recent years, including an aid agency food programme in Borno state last year where seven women were trampled to death.
Port Harcourt is the main oil hub in Nigeria, but despite the country’s oil wealth about 40% of Nigerians live below the poverty line.