No, what kills trees is applying the mulch too deeply. Freshly chipped mulch will tie up nitrogen, at the interface of soil and mulch, so you can, if you want, scatter a little high-nitrogen fertilizer first. It's not a major problem for a mature tree, so you can also ignore it. Apply the mulch no more than 4" deep, as widely as you can, and put no mulch in the 3-4" next to the trunk. You want the root-flare to be exposed, and mulch right next to the base of the tree can encourage rot in years to come.