Place the water in a medium pot and bring to a simmer over high heat. Remove from heat, add the tea bags and mint, cover, and let steep for 10 minutes.
Once the tea has steeped, discard the tea bags, add the sugar, and stir until the sugar has completely dissolved. Let cool to room temperature, then place in the refrigerator to cool completely, about 2 hours.
Once the tea is cold, remove the mint and stir in the lime juice. Taste and adjust with more lime juice or sugar as desired. Serve over ice with a sprig of mint and a wedge of lime (if using).