It depends on the pattern, and depends on how you are joining blocks together. If you are making the Temperature Quilt pattern, you can remove the papers once all the sides have been sewn.
If you are making National Park blocks, you may see that I don’t baste all the edges of the outside pieces. When I sew my NP blocks by EPP, I don’t baste the outer edge…this allows me to join the blocks by machine. This means that I want to have the exterior edges flat. I’ve noticed if I baste all the sides, I end up with a pucker, and the block doesn’t lay flat. So for NP blocks, I remove papers after the block is trimmed and starched.
I remove them when I’m done with the block, and before I starch. I really prefer starching over keeping the paper attached to the block. Who wants to remove all those little pieces of paper once they are sewn in an entire quilt?! Not I! For the park signs, I remove those after I have joined the park to the sign…but in general, I remove them when the block is done.