Put tea bags, cinnamon, cloves, allspice berries, ginger and brown sugar in a large heatproof container. Pour 4 cups of boiling water over tea and spices. Stir well. Let steep until liquid has cooled to room temperature, 2 hours. Strain into a pitcher; add cider and juice. Discard tea bags and spices.
If serving cold, place covered pitcher in refrigerator, chill until cold and serve over ice. If serving warm, pour into a slow cooker, cover and cook on low until warm, about 2 hours. Add rum to cooker, if desired, or add 1 Tbsp. rum each to individual glasses. Alternately, heat spiced cider in a large pot over low heat until warm and serve.