We present the results of simultaneous 450 μm and 850 μm polarization observations toward the massive star-forming region NGC 2071IR, a target of the BISTRO (B-fields in STar-forming Region Observations) Survey, using the POL-2 polarimeter and SCUBA-2 camera mounted on the James Clerk Maxwell Telescope. We find a pinched magnetic field morphology in the central dense core region, which could be due to a rotating toroidal disklike structure and a bipolar outflow originating from the central young stellar object IRS 3. Using the modified Davis–Chandrasekhar–Fermi method, we obtain a plane-of-sky magnetic field strength of 563 ± 421 μG in the central ∼0.12 pc region from 850 μm polarization data. The corresponding magnetic energy density of 2.04 × 10 -8 erg cm -3 is comparable to the turbulent and gravitational energy densities in the region. We find that the magnetic field direction is very well aligned with the whole of the IRS 3 bipolar outflow structure. We find that the median value of polarization fractions is 3.0% at 450 μm in the central 3′ region, which is larger than the median value of 1.2% at 850 μm. The trend could be due to the better alignment of warmer dust in the strong radiation environment. We also find that polarization fractions decrease with intensity at both wavelengths, with slopes, determined by fitting a Rician noise model of 0.59 ± 0.03 at 450 μm and 0.36 ± 0.04 at 850 μm, respectively. We think that the shallow slope at 850 μm is due to grain alignment at the center being assisted by strong radiation from the central young stellar objects.