Persimmons are a kind of fruit, and since fruit falls into the plant category, they can absolutely be included as part of a Paleo lifestyle. Just be sure not to overindulge.
Plants and Phytochemicals
Persimmons are rich in vitamin C and iron, as well as phytochemicals. Phytochemicals exist in plant foods to protect the plant, but they also convey significant health benefits for us when we eat them. They are powerful antioxidants and may even convey some cancer-fighting benefits.
Paleo and Carbohydrates
Despite their healthy vitamin content, persimmons are also fairly high in carbohydrates. While some people thrive on a diet consisting of around 100-150 grams of carbs, or even more, others find that extreme carbohydrate restriction works best for them. The latter will want to avoid fruit, but the only way to know what works best for you, personally, is to experiment and see what makes you feel your best.
Persimmons can be eaten fresh or dried. If you decide to go with dried, Mark Sisson has a whole post on the pros and cons of dried fruits, as well as instructions on how to dry them yourself.