The IR may be in a very narrow band. The visible wavelengths have different colors because there are a range of wavelengths that correspond to different cones in our eyes that roughly match red:green:blue sensors. If you shift the IR frequency up into the visible range, you would just get a luminance image (like grayscale) centered on one visible wavelength like red.

False color imaging sometimes applies colors to different luminance levels or sometimes it takes multiple images at different wavelengths and assigns RGB values to each of those wavelengths. The results are informative but require some editorial / aesthetic decisions to produce the best results.