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$_{⊙}$ 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$_⊙$ 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. ; Comment: Accepted to Astrophysical Journal