Can you see a rainbow with infrared goggles on?

No, you cannot see a rainbow with infrared goggles. Rainbows are formed by the refraction, dispersion, and reflection of visible light, which these goggles are not designed to detect. Infrared goggles primarily pick up infrared radiation, which is outside the visible light spectrum. Therefore, while you might see heat signatures or other infrared phenomena, the colors of a rainbow would not be visible.