Maybe in olden times people used to carry money in a small bag that they would seal with a metal ring. Actually when I was young I had something like this, I used to save pennies in it! When you purse your lips you are making a similar shape.

Hi Ken, Cool question. GuideDogSaint is right. I recently discussed this one with a student. We learned that "purse" in this context means: if you purse your lips, you bring them together tightly into a small circle, especially to show disapproval or doubt. (https://www.ldoceonline.com/dictionary/purse) When you look at photos of drawstring purses (small bags often used to carry money), that have been used for centuries, you will see why "purse" makes sense when we discuss the appearance of our lips. I'm adding a photo that I found on the following website: https://mydaintydreams.com/making-a-pirate-drawstring-coin-bag-pouch-free-template/ Also, I found the following information about the word: purse (v.) c. 1600 as "draw together and wrinkle" (as the strings of a money bag) For sense, compare pucker (v.) (https://www.etymonline.com/word/purse) I hope this helps.