Birch Bark can be harvested from birch trees.
To harvest, you simply right-click a birch tree and from the menu choose 'Peel Bark'. If you do not see such an option, you either clicked the wrong type of tree, or someone has already harvested all the bark before you.
Each tree can only yield 2 pieces of bark before being depleted. Assuming birch bark operates like Maple Leaves, the world will respawn the trees based on decay ticks. This needs to be confirmed.