The Kepler mission has yielded a large number of planet candidates from among the Kepler Objects of Interest (KOIs), but spectroscopic follow-up of these relatively faint stars is a serious bottleneck in confirming and characterizing these systems. We present motivation and survey design for an ongoing project with the Sloan Digital Sky Survey III multiplexed Apache Point Observatory Galactic Evolution Experiment (APOGEE) near-infrared spectrograph to monitor hundreds of KOI host stars. We report some of our first results using representative targets from our sample, which include current planet candidates that we find to be false positives, as well as candidates listed as false positives that we do not find to be spectroscopic binaries. With this survey, KOI hosts are observed over similar to 20 epochs at a radial velocity (RV) precision of 100-200 ms(-1). These observations can easily identify a majority of false positives caused by physically associated stellar or substellar binaries, and in many cases, fully characterize their orbits. We demonstrate that APOGEE is capable of achieving RV precision at the 100-200 ms(-1) level over long time baselines, and that APOGEE's multiplexing capability makes it substantially more efficient at identifying false positives due to binaries than other single-object spectrographs working to confirm KOIs as planets. These APOGEE RVs enable ancillary science projects, such as studies of fundamental stellar astrophysics or intrinsically rare substellar companions. The coadded APOGEE spectra can be used to derive stellar properties (T-eff, log g) and chemical abundances of over a dozen elements to probe correlations of planet properties with individual elemental abundances.