No, you don't.
Whether you need a Cinque Terre Train Card or not, depends on how many journeys by train between the villages you intend to make. The card costs €16 a day for an adult. A single train ticket costs €4 per journey if bought point-to-point.
Anyway, if you plan to use the train at least twice in a day and also want to hike one of the coastal paths, you better buy a Cinque Terre Train Card. Another benefit of this card is avoiding long lines to purchase train tickets during the crowded summer season.
Fore more information: Cinque Terre Card