The most long-lived class of nuclear waste, transuranics, is produced by neutron activation of Uranium-238 (~95% by mass) found in the low-enriched uranium used in power reactors. If you remove these nuclides from spent fuel by chemical separation and place them either into a Thorium (Uranium-233, really) reactor or a highly-enriched U-235 reactor, the transuranics will continue to absorb neutrons until they reach mass numbers where they spontaneously fission, and, since there is no U-238, there won't be any new transuranics.
Something of this nature is usually what is meant when people say "burn nuclear waste".
There will still be the other kind of nuclear waste, namely fission products, but their half-lives are on the order of decades instead of centuries.