He will immediately be replaced by co-founder and chief operating officer Paul Keary.
In a statement, Teneo said: “We want to thank Declan for his leadership and dedication over the past ten years in building Teneo into the world’s preeminent CEO advisory firm."
Pressure had mounted on Kelly to resign after one of Teneo's clients, General Motors, severed ties with the consultancy.
He had already relinquished some of his responsibilities at the firm after admitting to acting inappropriately while drunk at a charity concert chaired by Prince Harry and Meghan Markle.
The incident led Declan Kelly to be removed from the board of anti-poverty charity Global Citizen, which hosted Vax Live: The Concert to Reunite the World on 2 May.
A spokesperson for Kelly told the Financial Times: “At a large cocktail party that evening attended by dozens of people including Global Citizen board members and guests, Mr Kelly became inebriated and behaved inappropriately towards some women and men at the event."
They said Kelly “deeply regrets his actions” at the concert, which featured performances by Foo Fighters and Jennifer Lopez, and has apologised to those affected. He is "committed to sobriety" and was "undertaking ongoing counseling from healthcare professionals", the spokesperson added.
Kelly co-founded Teneo 10 years ago with Doug Band and Keary. It has grown into one of the largest corporate comms consultancies with 30 offices globally.
PRWeek estimates the UK business generated revenue of £48m in 2020. The company announced a new UK leadership team last summer, and earlier this year acquired the restructuring services business of Deloitte UK.
Teneo sold a majority stake to UK-based CVC Capital Partners in 2019, as former majority owner BC Partners exited after four-and-a-half years.