Process pineapple, sugar, and lime juice in a blender or food processor 30 seconds or until smooth, stopping to scrape down sides as needed. Pour through a fine wire-mesh strainer into a large pitcher, pressing to release juices. Discard solids. Bring water to a boil in a medium saucepan over medium-high heat. Remove from heat, and add tea bags. Cover and steep 7 minutes. Discard tea bags. Stir tea into pineapple mixture; cover and chill 4 to 6 hours. Gently stir chilled lemon-lime soft drink and rum into pineapple mixture just before serving. Place pineapple sherbet in a punch bowl or individual glasses. Pour pineapple mixture over sherbet. Serve immediately.