To press (one's lips) in and together so that they protrude.
1901, Matilde Serao, The Land of Cockayne, translator not credited, London: Heinemann, Chapter IV, p. 72, https://archive.org/details/landofcockayne00seraiala
The serving Sister pursed up her lips to remind him of the cloistral rule, almost as if she wanted to prevent any conversation between him and the nun.
1916, Leonid Andreyev, "An Original" in The Little Angel and Other Stories, translated by W. H. Lowe, New York: Alfred Knopf, p. 85, https://archive.org/details/littleangelother00andriala
Anton Ivanovich pursed up his lower lip so that his grey moustache pressed against the tip of his red pitted nose, took in all the officials with his rounded eyes, and after an unavoidable pause emitted a fat unctuous laugh.