Ok well maybe the title is a bit misleading. There are many, many places that you can set up a tripod in New York City and no authority will bother you. However these places rarely offer good vantage points. Pretty much every worthwhile landmark has tight security and the no-tripod rule is in strict enforcement. Tripods are not allowed in Times Square, Rockefeller Center, financial district, etc.
The other day while trying to photograph the Brooklyn Bridge from what I thought was an inconspicuous location, I was stopped by a police officer and threatened with a ticket. Here is a PDF from the Municipal Government explaining some laws regarding filming and photography on NYC streets: permits regarding tripods
The reasons are simple, tripods cause congestion on narrow sidewalks and streets. But as a photographer, how is one suppose to get a good photo in the city during darker hours?
Be sneaky, of course. The best time of day is early morning and late evening. I personally always carry my tripod retracted into a monopod, because monopods are generally accepted. Be casual, but be aware of who is around you. Police officers? Security guards? Don't spread that tripod. Mostly empty early morning streets with an occasional passerby? Then set up your tripod quickly and discreetly. Use your time within reason.
Being stopped by a police officer for something as innocuous as a tripod can be frustrating, but always be cooperative. Being difficult is a sure way to be slapped with a ticket.
Here are some shots I was lucky enough to capture with a tripod in no-tripod-zones.
ETA: One last thing... Brooklyn and Jersey's coast are generally very favorable and you can get all those classic shots of the cityscape from there at night. It's mostly Manhattan with such strict enforcement.
Good luck shooting!