Usually not for the shoot itself. The
New York City Mayor's Office of Media and Entertainment says the city only requires a film permit for shoots on city property, or ones using equipment beyond a handheld camera and tripod, asserting exclusive use of city property, or needing parking for production vehicles. An interior shoot inside a private rented studio generally isn't a city-permit situation, though your building or host may still have rules.
Insurance follows the same logic. If a production does need a city permit for something like spilling onto the sidewalk, the
Mayor's Office requires commercial general liability coverage of at least $1,000,000 per occurrence, naming the city as additional insured. That $1M rule ties to city permits, not private bookings, and studios set their own insurance terms.
Pro tip: Booking a private studio through Peerspace skips the public-space permit runaround you'd hit shooting on NYC streets, parks, or sidewalks, a real perk of a by-the-hour indoor space.