Damping of magnetic fields via ambipolar diffusion and decay of magnetohydrodynamical (MHD) turbulence in the post decoupling era heats the intergalactic medium (IGM). Collisional ionization weakly ionizes the IGM, producing an optical depth to scattering of the cosmic microwave background (CMB). The optical depth generated at $z\gg 10$ does not affect the "reionization bump" of the CMB polarization power spectrum at low multipoles, but affects the temperature and polarization power spectra at high multipoles. Using the Planck 2013 temperature and lensing data together with the WMAP 9-year polarization data, we constrain the present-day field strength, $B_0$, smoothed over the damping length at the decoupling epoch as a function of the spectral index, $n_B$. We find the 95% upper bounds of $B_0<0.56$, 0.31, and 0.14 nG for $n_B=-2.9$, $-2.5$, and $-1.5$, respectively. For these spectral indices, the optical depth is dominated by dissipation of the decaying MHD turbulence that occurs shortly after the decoupling epoch. Our limits are an order-of-magnitude stronger than the previous limits ignoring the effects of the fields on ionization history. Inverse Compton scattering of CMB photons off electrons in the heated IGM distorts the thermal spectrum of CMB. Our limits on $B_0$ imply that the $y$-type distortion from dissipation of fields in the post decoupling era should be smaller than $3\times 10^{-9}$, $10^{-9}$, and $2\times 10^{-10}$ for $n_B=-2.9$, $-2.5$, and $-1.5$, respectively.
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