In "All toric l.c.i.-singularities admit projective crepant resolutions", the authors say (second page)
Families of Gorenstein non-l.c.i. toric singularities that have such special full resolutions seem to exist only rarely.
I would like to think they would not say this if there were no examples of Gorenstein non-l.c.i. toric singularities.
Just to emphasize what everyone else is saying, being l.c.i. is an intrinsic and local condition on a ring. So the natural question is to ask about an affine toric variety, since the question for a projective toric varitey just comes down to checking in every coordinate chart. And l.c.i. implies Gorenstein, which not all toric varieties are, so you at least want to include the condition that your toric variety be Gorenstein.