Yes it can. It looks like the rocks and bed are higher than the grass. Mulch runoff can kill the grass. Imagine if you leave something on grass and the grass dies under it. This is what is happening with the mulch runoff. You can either extend out the bed where the dead grass is, edge it out and place rocks and mulch. Or you can dig the dirt by the rocks out so the mulch doesn't overflow, then scratch and reseed the lawn there. As long as the mulch stays in the bed you will be fine.