  Spatially fluctuating primordial magnetic fields (PMFs) inhomogeneously
reheat the Universe when they dissipate deep inside the horizon before
recombination. Such an energy injection turns into an additional photon
temperature perturbation. We investigate secondary cosmic microwave background
(CMB) temperature anisotropies originated from this mechanism, which we call
{\it inhomogeneous magnetic reheating}. We find that it can bring us
information about non-linear coupling between PMFs and primordial curvature
perturbations parametrized by $b_{\rm NL}$, which should be important for
probing the generation mechanism of PMFs. In fact, by using current CMB
observations, we obtain an upper bound on the non-linear parameter as $\log
(b_{\rm NL} (B_{\lambda}/{\rm nG})^2) \lesssim {-36.5n_{B} - 94.0}$ with
$B_{\lambda}$ and $n_{\rm B}$ being a magnetic field amplitude smoothed over
$\lambda=1\; {\rm Mpc}$ scale and a spectral index of the PMF power spectrum,
respectively. Our constraints are far stronger than a previous forecast based
on the future CMB spectral distortion anisotropy measurements because
inhomogeneous magnetic reheating covers a much wider range of scales, i.e.,
$1\; {\rm Mpc}^{-1} \lesssim k\lesssim 10^{15}\; {\rm Mpc}^{-1}$.
