Under their tenure the theatre has undergone a Â£22m redevelopment and doubled audience numbers, plus several transfers to the West End including Enron and Sweeney Todd.
They joined the theatre in 2005 and oversaw its 50th anniversary in 2012.
Theatre chairman Sir William Castell described them as "inspiring leaders".
Under Church and Finch, the company has also staged more than 100 productions including 22 new plays and has won more than 40 awards. Audiences played to 95% capacity in 2014.
They have also presided over 48 productions which have transferred to the West End or secured international and UK tours. These include Macbeth with Patrick Stewart, Singin' in the Rain, Private Lives, King Lear with Frank Langella, The Pajama Game and, more recently, Church's production of Taken at Midnight and Gypsy, directed by Jonathan Kent.
Church described being a "custodian" of the theatre as a "privilege and a joy", adding that he had "worked alongside some of the country's most skilled actors and creative teams".
"I hope, between us, we have helped contribute to the current strength of British theatre," he said, stressing the "important role of subsidy in creating vibrant regional arts organisations".
He said it would be "painful leaving a theatre that feels like our creative home" but added that regional theatres "thrive on new ideas and new energies".
Finch hailed their "loyal audiences, generous supporters and sponsors", saying that without their support they would "never have been able to achieve what we have". He said it was "the right time for us to pursue new opportunities".
The process to appoint a new artistic director will begin in August and a successor is expected to be announced at the end of this year.
The new executive director will be appointed when Church's successor is in place.
Chichester Festival Theatre, which has a distinctive Grade Two* listed hexagonal structure, is one of the largest in southern England. It first opened its doors in 1962 under the artistic leadership of Laurence Olivier.
Chichester Festival Theatre is to lose both its artistic director Jonathan Church and executive director Alan Finch at the end of September 2016.