Menzies said Mr Dollman would step down at the annual meeting, subject to a successor being in place, to concentrate on building a portfolio career.
The announcement came as the Edinburgh-based logistics group said it was closing its cargo-handling operations in Chicago – the latest in a series of restructuring moves that had been piloted by Mr Dollman – and that its year-end debt would be less than £100 million.
Menzies said the group had continued to make positive progress and the board anticipated the full-year result to be in line with its expectations.
It went on: "Menzies Aviation has decided to close its cargo- handling operations at Chicago Airport, having exhausted all alternative options over the past two years."
It said this move would result in an exceptional charge of £7m in 2012 but it would improve earnings before tax and interest in 2013 by around £1.4m.
"Following the series of restructuring actions across both divisions, the full-year exceptional charge will be in the region of £18m.
"The enhancements to underlying earnings from these actions are included within the board's expectations and leave the group in a stronger position with a more stable platform for growth."
It said the group interest charge would rise by around £2.5m in 2013, but that this was purely due to IAS19 accounting rules.
Four years ago, with debt at £200m, John Menzies saw its market value crash by more than 90% in a year to £26m, hit by the double crisis in banking and aviation.
Mr Dollman, who had joined as finance director in 2002 from the same role at William Grant, led the rescue operation of the group which in 2007 had dispensed with a chief executive in favour of a triumvirate of two operational directors and himself.
He renegotiated £45m of debt and oversaw the cost-cutting programme, which saw Menzies exit unprofitable operations in its aviation division. This year saw a withdrawal from cargo- handling at four UK airports, including Glasgow, where employees transferred to new operator Servisair.
Last month Mr Dollman told the City that Menzies had taken on another £25m of debt to target bolt-on acquisitions, after a year where profits (in constant currency) had beaten expectations and in which the shares had topped 650p – some 14 times their nadir in March 2009.
Chairman Iain Napier said: "Paul has done an outstanding job for John Menzies plc over the past 10 years and we wish him well for the future.
"He has been an integral part of the executive team and has played a key role in the success of the group. A full search process is underway and we will make a further announcement at the appropriate time."
The group also announced the departure from the board in May of Ian Harrison, one of the long-standing non-executive directors connected to the founding Menzies family who has been on the board since 1987.
Mr Napier acknowledged Mr Harrison's contribution to the group.