We present a template-fitting algorithm for determining photometric redshifts, z phot, of candidate high-redshift gamma-ray bursts (GRBs). Using afterglow photometry, obtained by the Reionization and Transients InfraRed (RATIR) camera, this algorithm accounts for the intrinsic GRB afterglow spectral energy distribution, host dust extinction, and the effect of neutral hydrogen (local and cosmological) along the line of sight. We present the results obtained by this algorithm and the RATIR photometry of GRB 130606A, finding a range of best-fit solutions, 5.6 < z _phot < 6.0, for models of several host dust extinction laws (none, the Milky Way, Large Magellanic Clouds, and Small Magellanic Clouds), consistent with spectroscopic measurements of the redshift of this GRB. Using simulated RATIR photometry, we find that our algorithm provides precise measures of z_phot in the ranges of 4 < z_phot lesssim 8 and 9 < z_phot < 10 and can robustly determine when z _phot > 4. Further testing highlights the required caution in cases of highly dust-extincted host galaxies. These tests also show that our algorithm does not erroneously find z _phot < 4 when z _sim > 4, thereby minimizing false negatives and allowing us to rapidly identify all potential high-redshift events.