A reader from outside the United States wrote to me to ask this question. It has a simple-enough answer -- no, you do not need to speak a single word of French to visit New Orleans -- but it brings up some interesting perspectives about the city, especially as seen through the eyes of those who may not be from the region.
New Orleans was a historically French city, a major trading outpost in the Louisiana colony that was populated with French merchants and their families, sailors, traders, and a large number of Francophone enslaved people (and later, free people of color who also spoke French).
Over the years, the colony changed hands several times, becoming a Spanish colony for a good stretch (during which it was still primarily a Francophone region, at least for social affairs, if not legal ones), and then reverting back to the French before the Louisiana Purchase made it an American possession.
At this point, an influx of English-speaking Americans flooded the city, widely expanding both its population and its footprint. "Les Américains" settled upriver from Canal Street in what is now the Central Business District, the Warehouse District, and the iconic Garden District. The French-speakers, known as Créoles, remained in the French Quarter and downriver (with people of color heavily populating the Tremé, the oldest black neighborhood in the United States).
At this point, the city was certainly bilingual. Creoles did their business in French, Americans did their business in English, and reluctant translations took place when the two populations met (something, it seems, they preferred to do as little as possible for quite some time).
Eventually, and particularly in the post-Civil War period, French began to give way and English began to dominate, though many French colloquialisms (and those from other languages spoken by immigrant groups like the Italians and Germans) passed into English, as did a distinctive accent that is markedly different from those elsewhere in the Deep South.
By WWI, English was the primary language for nearly everyone (except, perhaps, recent immigrants and a handful of stubborn older Creoles), and in the modern era, there are essentially no native speakers of traditional New Orleans Creole French. (Certainly some families remain bilingual by choice).
Some French speakers may actually find Louisianans' pronunciation of some of their French vocabulary rather baffling and occasionally even aggravating. One of the more popular Mardi Gras krewes, for example, is called "Krewe du Vieux" -- a French-speaker will want to say the final word as "vyuh," but in New Orleans, it's pronounced "voo." "Beaucoup" is typically pronounced "bookoo," and the wide variety of street names present a unique challenge that even many transplants admit to having never figured out.
Pro tip: if you do speak French, don't bother over-pronouncing things in the "correct" way. It's a very obvious mark of a tourist and a bit grating. Listen and repeat pronunciations you hear; it's really okay.
A side note: though New Orleans French is largely a relic at this point, there are still native speakers of Louisiana varieties of French throughout the state, primarily in the triangle known as Acadiana, where many Cajun and Creole families have maintained their unique language variants despite outside pressure over the years. Still, nearly everyone but a handful of very elderly folks are bilingual, but native speakers do, indeed, exist. Many children now study in public French Immersion schools, as well.
These rural French dialects are a few hundred years removed from standard French, but are mutually intelligible (despite what a few particularly snooty folks from both sides of the pond will tell you) -- friendly folks from France, Canada, French Louisiana, and other Francophone regions have no significant trouble conversing.
