Bring 3 cups of water to a boil. Remove from heat and stir in the sugar and a pinch of salt. Add the raspberries and the tea bags. Steep the tea bags for 5 minutes; remove the bags and discard.
Refrigerate the tea until cool, 1 to 2 hours. Stir in the lime juice.
Pour the mixture into 10 3-ounce or 8 4-ounce ice-pop molds, dividing evenly. Insert the ice-pop sticks and freeze until firm, at least 4 hours and up to 4 days.