@mathmomike has posted his 2016 predictions:
- Improved methods for using phylogenies to study processes on trees (including the evolution of morphological and continuous characters).
- Development of techniques for updating/refining/enlarging existing phylogenies as new data are added.
- Development of phylogenetic methods based on models that allow site-specific rates and processes that change across a tree.
- Statistical methods for generating random phylogenetic networks and inferring networks based on the trees they display.
- Improved techniques for posterior estimation of large trees, and further development of invariant-based phylogeny reconstruction methods.
You can see a discussion of his 2011 predictions here.