If you go to the first link that Bmac has provided, it might bear noting that: 1) this summary reflects the final rule published on 6/11/10 (see"History"), 2) that the discussion was updated as recently as 11/1/11 (see the bolded paragraph under "When A Permit Is Required"), and 3) the webpage carries a 2012 copyright notice (bottom left). Conclusion: Barring additional pursuasive evidence to the contrary, this is current policy, at least as far as the Mayor's Office is concerned.

Re the substance of the policy: However inartfully expressed, it's clear that an individual, merely using a tripod in order to "film", does not require a permit under the MOFTB regulation to photograph in NYC unless the individual is asserting, or intends to assert, exclusive use of City property while using a tripod. This is expressed in several ways in the policy statement.

Other issue: What other NYC agencies, if any, might have jurisdiction over issuing permits that could be relevant to the use of a tripod for photographic purposes? If none, then no permit needed.

Note for the OP: You can apply for an optional permit at no cost to you. You could send them a list that includes all of the sites listed by others and as many dates as you might want to shoot. Might cover you if the gendarmes hassle you.

Thanks, Bmac, for the links.

Greenwich Village - I'd shoot Washington Square Arch.
Little Italy I'd shoot a scene on Mulberry Street taking in the many Italian Restaurants
The Financial District you have to shoot the Bull Sculpture at the south end of Broadway.
Lower East side you can shoot Delancy Street especially on a Sunday
Midtown - Gotta be the Empire State Building or Grand Central Station.
Upper West Side try the Museum of Natural History
Upper East Side - Central Park/Central Park Zoo
Hells Kitchen - The Intrepid Air/Space Museum
Meat Packing District - Highline Park (Park built on Elevated Railroad Tracks)
World Trade Center Memorial a must
Times Square - Especially at night
Rockefeller Center Ice Rink
The Guggenheim - Round Building that's really unique.
Don't forget Hotels - Waldorf - Magnificant Lobby, The Plaza Incredable.
The Metropolitan Museum and Grounds You'll be there all day
There ya go. That should keep you busy for a while.

You can take the double decker tour bus to most areas, on and off all day and night one fee. Don't forget the bridges, south street sea port, the parks, Elevated park on west side between 10th and 30th, china town[ day time ], the ferry, 42nd st., wall street from one end looking up the street, the churches , pop into hotels and check with consignore [ spell check please ] their suggestions for immediate area.
And yes you can use tripod . Don't forget to look up at buildings, amazing shots up there.
Okay, thats all i got, have a great time doing it, don't make it a job. Vinniepics