  We present the analysis of supernova remnants (SNRs) in the Large Magellanic
Cloud (LMC) and their influence on the environment at far-infrared (FIR) and
submillimeter wavelengths. We use new observations obtained with the {\it
Herschel} Space Observatory and archival data obtained with the {\it Spitzer}
Space Telescope, to make the first FIR atlas of these objects. The SNRs are not
clearly discernible at FIR wavelengths, however their influence becomes
apparent in maps of dust mass and dust temperature, which we constructed by
fitting a modified black-body to the observed spectral energy distribution in
each sightline. Most of the dust that is seen is pre-existing interstellar dust
in which SNRs leave imprints. The temperature maps clearly reveal SNRs heating
surrounding dust, while the mass maps indicate the removal of
3.7$^{+7.5}_{-2.5}$ M$_{\odot}$ of dust per SNR. This agrees with the
calculations by others that significant amounts of dust are sputtered by SNRs.
Under the assumption that dust is sputtered and not merely pushed away, we
estimate a dust destruction rate in the LMC of $0.037^{+0.075}_{-0.025}$
M$_\odot$ yr$^{-1}$ due to SNRs, yielding an average lifetime for interstellar
dust of $2^{+4.0}_{-1.3}\times10^7$ yr. We conclude that sputtering of dust by
SNRs may be an important ingredient in models of galactic evolution, that
supernovae may destroy more dust than they produce, and that they therefore may
not be net producers of long lived dust in galaxies.
