Public notices in NYC occasionally use slang or colloquial language. “Don’t do NYC dirty” isn’t grammatically correct. To “do” someone “dirty” usually means to be inconsiderate of someone, or to betray or harm them in worse cases. Example: 👨: hey did Robert give you back your phone charger? 👨🏾🦱: Nah, he did me dirty. (Robert forgot to return the phone charger.) This notice also says to “Trash your trash”. Trash means garbage, but it is also a verb meaning to throw something away. Example: 👨: hey, do you still have the receipt for the new phone charger? 👨🏾🦱: Nah, I trashed it.