There will be no direct replacements for their roles. The restructure comes after Emma Fraser, head of brand communications, was made redundant in June.
It is believed newspaper sales director Steve McLaughlin and commercial director Dave King will take on some marketing responsibilities.
A TMG spokesman says: “The changes will streamline operations to align digital and marketing activity more closely with the commercially focused areas of the business.”
Flanigan, who took over the role at the start of the year, replaced Katie Vanneck, who defected to Times Media in July last year. Flanigan was previously director of marketing and communications at AOL.
Moore’s departure comes less than six months after his promotion from the role of executive director of consumer markets.
It is not known if the cost-cutting measures will have any impact on TMG’s agency arrangements. In February, it appointed Carat to handle its £4m media account, while Adam & Eve and Amp were appointed the following month to handle its £5m advertising business.
Further changes are expected at the company, with Lorraine Twohill, Google’s vice-president of marketing for Europe, the Middle East and Africa, joining the TMG board in a non-executive role this week.