The cause of the problem hasn't been uncovered. In the meantime, Bungie advises those affected to call Xbox customer support (1-800-4-MY-XBOX) to report the issue. We've contacted Microsoft and will update this post when more information becomes available.
Update: Microsoft has responded, saying, "We are aware of a very small number of customers reporting that their copy of Halo: Reach is causing a "disc read" error." The company advises players to call customer support for "troubleshooting assistance."