Using the imaging polarimeter for the Submillimeter Common User Bolometric Array at the James Clerk Maxwell Telescope, we have detected polarized thermal emission at 850 μm from dust toward three star-forming core systems in the Orion B molecular cloud: NGC 2071, NGC 2024, and LBS 23N (HH 24). The polarization patterns are not indicative of those expected for magnetic fields dominated by a single field direction, and all exhibit diminished polarization percentages toward the highest intensity peaks. NGC 2024 has the most organized polarization pattern, which is centered consistently along the length of a chain of seven far-infrared sources. We have modeled NGC 2024 using a helical-field geometry threading a curved filament and also as a magnetic field swept up by the ionization front of the expanding H II region. In the latter case, the field is bent by the dense ridge, which accounts for both the polarization pattern and existing measurements of the line-of-sight field strength toward the northern cores FIR 1-4. The direction of the net magnetic field within NGC 2071 is perpendicular to the dominant outflow in that region. Despite evidence that line contamination exists in the 850 μm continuum, the levels of polarization measured indicate that the polarized emission is dominated by dust.