If I buy a new copy of Animal Crossing, will it remember my save data?
You can't resume playing on your current game progress. According to this nintendo-support site the game progress is saved on the cartridge itself.
While some Game Cards do save SpotPass and StreetPass information to the SD card or system memory, game progress (levels completed, items collected, etc.) is saved to the Nintendo 3DS Game Card.
Note that save data for titles downloaded through the Nintendo eShop (not played on a game card) is saved to the system memory.