Don’t say you’re “from New York” when you’re from New Jersey or Long Island. There are very nice parts of New Jersey and Long Island; some very nice people live there. But this is not Boston - you don’t get to say you’re “from New York City” if you’re from slightly outside it. If your prevarication is discovered, this is a quick route to contempt.
Never ever ever EVER refer to the city as “the Big Apple.” If you say this, you are a tourist, and a clueless one at that. Using the phrases “only in New York!” and “a New York minute” falls in the same category, but they may be used, sparingly, by long-time residents, with a heavy dose of irony.
Don’t refer to the subway lines by their color. Instead, refer to them by their numbers and letters - e.g. it’s not the “Green Line,” it’s the “4, 5, 6.” When referring to a specific service along that line, each is called a “train,” rather than a “subway” - e.g. the “6 train,” not the “6 subway.” When referring to the entire system, it’s the “subway” - not the “Metro,” the “Underground,” etc.
Don’t wear “I Heart NY” t-shirts, or indeed any article of clothing that mentions New York in any capacity, with the exception of gear supporting a sports team.
If there is a wait for something or a bottleneck, don’t mob it - form a line. And when a line has been formed do NOT try to cut it. Seriously. This is for your own health.
When you get on a bus or step up to a subway turnstile, have your change or MetroCard ready. There’s a special circle of hell devoted to people who waste 20 seconds of everyone else’s time with their fumbling.
Don’t ask people where you can find good “New York Pizza.” In New York, it’s just called pizza - most New Yorkers don’t even know “New York Pizza” is a thing outside New York, or that there is a “New York-style” (see Where can you get New York-style Pizza in London? and its ilk). Just go to the local corner pizza shop and help yourself; I promise it’ll have “New York-style pizza” unless it says very explicitly otherwise.
Corollary to the above - do not say you prefer Chicago, New Haven or (God help you) California pizza. This is a direct route to a heated argument.
I’ve seen more petty spats over this than over just about anything else in New York; don’t steal another’s cab. But how, you might ask, should one know a cab’s rightful owner? It’s simple. Taxi possession works on a territorial, first-come-first-serve system. If someone is trying to hail a cab on the street, he has established his territory there, so don’t infringe by trying to hail one there too. Going farther up the stream of traffic to cut him off is also taboo. The polite distance varies by location and time. In Midtown at rush hour and the Lower East at 2am, a block is standard; early mornings in residential neighborhoods can require two full blocks or more.
When you refer to locations in Manhattan, don’t give the Avenue first - always start with the Street. If you’re going to 9th Street and 3rd Avenue, say “Ninth and Third,” never “Third and Ninth.”
Perhaps less of a faux pas, but a sure tipoff that you’re a tourist; if you’re in Manhattan, don’t refer to “North” and “South;” it’s “Uptown” and “Downtown,” respectively.
If you are not front-office at a financial institution,don’t call yourself a “banker.” There’s absolutely nothing wrong with managing a Wells Fargo branch; there’s nothing wrong with working on the tech side at a financial institution. In New York, however, “banker” indicates that you’re a front-office type. If you’re caught cheating here, you’ll totally blow your credibility and possibly take some ridicule.
New York is a walking city. Very few places are located directly on public transit and most journeys require at least some walking. It’s often the fastest way around, and it’s definitely the healthiest, cheapest, and best for the environment. If you’re in New York, don’t complain about having to walk.
New York eats late - don’t propose dinner earlier than 7pm unless the other party has kids. People won’t hate you for violating this, but they may give you a strange look.
This may seem like a no-brainer, but I’ve encountered it way too often: if you are a tourist, don’t bring up 9/11. Out-of-towners are frequently more open and talkier than New Yorkers, and we appreciate that the rest of the country felt a kinship with us. Admittedly, sometimes, we do feel like talking about it. But bringing it up first, or in casual conversation, is just poor taste.
Perhaps less of a faux pas and more of a pet peeve; don’t ask “What’s a good restaurant?” or “What’s a good hotel?” There are literally thousands of restaurants and hotels in New York, many of them good. Specify what you’re looking for (price point, atmosphere, neighborhood or access to neighborhoods, type of cuisine, etc.) and you’ll get a much more positive response.
This one is absolutely vital - don’t interfere with others’ privacy. New York is a very crowded place. The way people deal with it is to create their own space. Thus, what outsiders often see as aloofness and isolation is, in fact, a sign of community; there is a shared ethos that everyone respects others’ privacy and expects others to respect his own. This is chiefly communicated through eye contact. If you stare at someone on the subway: if you linger in looking out your window into someone else’s bedroom; if you react to or interrupt a celebrity; or if you seem to be intentionally listening in to another’s conversation, you are violating one of New York’s most sacred unwritten rules. Keep yourself to yourself, buddy, and let others do the same.
Don’t tip like you do at home; tip at New York rates. This has already been touched on by a few other answers, but it needs some explanation. The people who make life easier for you in New York —taxi drivers, servers, etc.— are paid scant wages and depend on tips for a large part of their income. Yet they still have to survive in one of the most expensive cities in the world. If you fail to tip at New York rates for decent service, you are not paying for that service. Tipping 15% is an insult and 16-18% is parsimonious. While I hate to point fingers, I’m looking particularly at our friends from Europe here. If you think Americans’ being loud, fat, monolingual, and ignorant in your beloved cities is obnoxious, your failure to pay for service rendered is downright criminal. It’s an expensive city; pull the Gauloise from your lips, reach into the pocket of your lederhosen, pull out an extra quid or two and pony up!
Do not touch a stranger’s kid. This is a sad rule, because nearly all the people who break it are extraordinarily warm and sweet and have nothing but the best intentions. In a lot of places, children are raised communally; it may be normal to high five or pick up a stranger’s kid who walks up, to lift her onto an empty seat on the subway, to play patty-cakes with her, or to chastise her if she misbehaves. But do NOT try this in New York. While New York is one of the safest cities in America, parents of city kids are protective and will not be happy.
Don’t fake a New York accent. You’ll sound like an idiot, and most people here speak either with a foreign accent (if they speak English), or with a nondescript accent anyway.