A particle can only annihilate with its own antiparticle. So for example an electron will annihilate with a positron but not with an anti-muon or anti-proton.
So dark matter particles will only annihilate with anti-dark matter particles of the same type. If you were, for example, to place positrons or anti-protons and dark matter together nothing would happen.
As tfb says in a comment we don't know what dark matter is, but if it is a particle then we expect dark matter particles and dark matter antiparticles to annihilate with each other in the same way non-dark particles do. In fact this is only of the ways dark matter is being looked for. The annihilations should produce an excess of gamma rays above the expected backgrounds, and detecting such an excess would be strong evidence that dark matter particles exist. So far no such detections have been made, though we've had several false positives.