I suspect the answer is no. But I don't know enough to be sure that that is the case. Of course phylogenetic tree construction uses biological knowledge, e.g special distance metrics, but does it brings anything new at the level of clustering agorithm (e.g hierarchical, neighbor-joining etc).
One difference is that phylogenetic tree construction algorithms will typically try to jointly estimate the tree and the transition parameters. A general clustering algorithm might assume that a difference in one feature would make the same contribution to 'distance' between clusters as a difference in another feature. In phylogenetic tree construction, it is explicit that not all differences between taxa represent the same evolutionary distance between taxa, and the parameters governing the distribution of distance/difference may be jointly estimated with the tree.