What results are immediately generalised to higher dimensions, in light of Schoen and Yau's recent preprint?

I know two applications:
a) The convergence of the Yamabe flow in dimension 6 and higher
Simon Brendle, Invent. math. 170, 541–576 (2007)
DOI: 10.1007/s00222-007-0074-x
b) Solution of the equivariant Yamabe problem
Farid Madani, Hebey-Vaugon conjecture II. (English, French summary)
C. R. Math. Acad. Sci. Paris 350 (2012), no. 17-18, 849–852.

Both article use the positive mass theorem to construct a good test function for the Yamabe functional.

For application b) the history is a bit distorted and not easily visible, so I will give more explanation.

Let $G$ be a compact Lie group acting on a compact manifold $M$. The equivariant Yamabe problem is to show the following. If $g_0$ is a $G$-invariant Riemannian metric on $M$, let $[g_0]^G$ be the set of all $G$-invariant metrics of volume 1 conformal to $g_0$.

The equivariant Yamabe problem is to minimize the Einstein-Hilbert functional $$g\mapsto \int_M~ \mathrm{Scal}^g~ dv^g$$ among all metrics $g\in [g_0]$ with the constraint $\mathrm{vol}(M,g)=1$. It was considered as being solved by an article
E. Hebey, M. Vaugon, Le problème de Yamabe équivariant, Bull. Sci. Math. 117 (1993) 241–286.
however this article used Schoen's Weyl vanishing conjecture which turned out not to hold in large dimensions (Counterexample by Brendle). Madani had developed in the above reference an alternative proof, closer to the standard proof of the classical Yamabe problem. In some cases he uses the positive mass theorem to construct a good test function. With the positive mass theorem in general, the solution of the equivariant Yamabe problem follows in full generality.

Probably Brendle's test function from the publication in inventiones could have used for the same purpose, but I do not know a reference where it was worked out how to use Brendle's test function to solve the equivariant Yamabe problem.

The progress of Schoen and Yau also allows to generalize many statements previously known only for spin manifolds to non-spin manifolds. See e.g. a preprint by Thomas Schick and Simone Cecchini, arXiv math.GT 1810.02116. In particular, they are now able to prove the following: An enlargeable metric (complete or not) cannot have uniformly positive scalar curvature. Therefore, a closed enlargeable manifold cannot carry any metric of positive scalar curvature.

The compactness of solutions to the Yamabe problem (and its version with boundary) holds upto dimension 24 if the PMT is true. (For higher dimensions there are counterexamples.)