Combine persimmons with sugar in blender or food processor and process on high speed until very smooth, about 30 seconds. Pour through a strainer to measure out 2 cups of purée, reserving remainder for another use. Transfer purée to a medium-sized mixing bowl and whisk in tea, lemon juice, and salt to taste. Chill in freezer until mixture is very cold, 2 to 3 hours. Churn in ice cream maker according to manufacturer's instructions. Serve immediately as soft serve or transfer to an airtight container and chill in freezer for 4 hours for a firmer texture.
