MONTAUK, N.Y.—For a dozen years, Apollo Global Management CEO Marc Rowan has operated a side gig in Montauk: a seafood shack that draws celebrities and influencers with sunset views, Dom Pérignon and a $97 lobster salad. Just don’t call it a restaurant in front of the neighbors. Many of them say it shouldn’t be there at all. They contend the business, known as Duryea’s Lobster Deck, has never gone through the proper approval process. “Money and power should not exempt you from the same laws that we have to follow,” said Hannah Kirschner, who’s had a home nearby since 1994.…

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