Does Gríma Wormtongue die in the LOTR?

Yes. Although I'm not sure on the literature part, but in the movies (extended edition) he meets his end. How? When the fellowship meet up at thrashed and flooded region of Isengard and it's tower, Orthanc, Gandalf and the others ask Saruman for Sauron's next move. After Gandalf breaks Saruman's staff, Theoden notices Grima Wormtongue when he comes up from behind Saruman atop the tower and reminds him that he was a man of Rohan, pleading him to come down and be free of Saruman's control. Saruman strikes at Grima, angering him. Just as Saruman agrees to provide the information, Grima takes out his knife and plunges it into Saruman's back. Legolas immediately pulls an arrow and pierces Grima's heart as Saruman plummets to the ground and gets impaled by a spike on a wheel. Hope I helped.