Spectrally-resolved imaging provides a spectrum for each pixel of an image that, in the mid-infrared, can enable its chemical composition to be mapped by exploiting the correlation between spectroscopic features and specific molecular groups. The compatibility of Fourier-transform interferometry with full-field imaging makes it the spectroscopic method of choice, but Nyquist-limited fringe sampling restricts the increments of the interferometer arm length to no more than a few microns, making the acquisition time-consuming. Here, we demonstrate a compressive hyperspectral imaging strategy that combines non-uniform sampling and a smoothness-promoting prior to acquire data at 15% of the Nyquist rate, providing a significant acquisition-rate improvement over state-of-the-art techniques. By illuminating test objects with a sequence of suitably designed light spectra, we demonstrate compressive hyperspectral imaging across the 700-1400 cm-1 region in transmission mode. A post-processing analysis of the resulting hyperspectral images shows the potential of the method for efficient non-destructive classification of different materials on painted cultural heritage.