Place tea bags and mint sprigs in a large heat-proof measuring cup. Pour boiling water over tea bag mixture and cover measuring cup with a small plate. Steep tea until desired strength is reached, 15 to 20 minutes. Pour steeped tea through a strainer into a large mason jar, squeezing excess liquid from tea bags. Discard mint and tea bags.

Stir sugar into tea until dissolved; add lemon juice, lime juice, and vanilla extract. Stir cold water into tea mixture. Place lid on the jar and refrigerate until completely chilled, 2 hours to overnight.

Fill a punch bowl or drink dispenser with ice and add tea.