  The thermal Sunyaev-Zel'dovich (tSZ) effect is the distortion generated in
the cosmic microwave background (CMB) spectrum by the inverse-Compton
scattering of CMB photons off free, energetic electrons, primarily located in
the intracluster medium (ICM). Cosmic infrared background (CIB) photons from
thermal dust emission in star-forming galaxies are expected to undergo the same
process. In this work, we perform the first calculation of the resulting
tSZ-like distortion in the CIB. Focusing on the CIB monopole, we use a halo
model approach to calculate both the CIB signal and the Compton-$y$ field that
generates the distortion. We self-consistently account for the redshift
co-evolution of the CIB and Compton-$y$ fields: they are (partially) sourced by
the same dark matter halos, which introduces new aspects to the calculation as
compared to the CMB case. We find that the inverse-Compton distortion to the
CIB monopole spectrum has a positive (negative) peak amplitude of $\approx 4$
Jy/sr ($\approx -5$ Jy/sr) at 2260 GHz (940 GHz). In contrast to the usual tSZ
effect, the distortion to the CIB spectrum has two null frequencies, at
approximately 196 GHz and 1490 GHz. We perform a Fisher matrix calculation to
forecast the detectability of this new distortion signal by future experiments.
$\textit{PIXIE}$ would have sufficient instrumental sensitivity to detect the
signal at $4\sigma$, but foreground contamination reduces the projected
signal-to-noise by a factor of $\approx 70$. A future ESA Voyage 2050
spectrometer could detect the CIB distortion at $\approx 5\sigma$ significance,
even after marginalizing over foregrounds. A measurement of this signal would
provide new information on the star formation history of the Universe, and the
distortion anisotropies may be accessible by near-future ground-based
experiments.
