"chalky, sticky mineral used in making mortar," from Old English lim "sticky substance, birdlime;" also "mortar, cement, gluten," from Proto-Germanic *leimaz (source also of Old Saxon, Old Norse, Danish lim, Dutch lijm, German Leim "birdlime"), from PIE root *(s)lei- "slime, slimy, sticky" (source also of Latin limus "slime, mud, mire," linere "to smear;" see slime (n.)).
Bird-lime is prepared from the bark of the holly, it was spread on twigs and used for catching small birds. The lime used in building, etc. is made by putting limestone or shells in a red heat, which burns off the carbonic acid and leaves a brittle white solid which dissolves easily in water. Hence lime-kiln (late 13c.), lime-burner (early 14c.). As a verb, c. 1200, from the noun.