It looks like they won't have to pay if they agree to use only the top or bottom of the bridge (and not both like they have since 1988)
NY Times wrote:
The M.T.A. has quietly demanded roughly $750,000 a year from the organization that runs the marathon, to make up for the toll revenue that the authority loses when it closes the Verrazzano — North America’s longest suspension bridge — to vehicular traffic, according to interviews and memos reviewed by The New York Times.
The organization, the New York Road Runners, has yet to acquiesce, prompting the M.T.A. to play hardball. The authority initially threatened to restrict runners to the bridge’s shadowy lower deck during the 26.2-mile race in November. But in recent weeks, the M.T.A. slightly relented and said the race could use the upper level if the Road Runners preferred it to the lower level.
Still, the authority said there was no way it would allow both decks to be used without a payment agreement in hand. The marathon has used both decks since 1988.
https://www.nytimes.com/2024/04/03/nyregion/marathon-tolls-mta.html
