One place is a paper of Kovacs: Irrational Centers[Irrational Centers][1]
In particular, the associated primes/points of the higher direct images are defined to be "irrational centers", which can then be used to obtain depth estimates. In the case you are interested in (ie no boundary case) then this was looked at previously by Alexeev-Hacon in an unpublished preprint (hopefully this will be available soon).
Is there something in particular you are hoping for?
There are related things for non-resolution of singularities statements, but just (flat) maps $f : Y \to D$. People have also looked at higher direct images of $\mathcal{O}_Y$ keeping in mind the singularities of the fibers (this is the reason for the original interest in Du Bois/DB singularities).
Finally, if you take $Y$ to be the reduced pre-image of $D$ (instead of the strict transform), then one can also look at the higher direct images, and now you are getting close to things again near Du Bois singularities. I wonder if seminormality can be detected similar to normality above...
EDIT: Based on more details from the questioner, let me answer the question. Suppose that $X$ is a $d$-dimensional Cohen-Macaulay variety which has rational singularities except at a single point $x \in X$. It follows from Kovacs' Lemma 3.3 cited below that for a resolution of singularties $f : X' \to X$, we have $R^i f_* O_{X'} = 0$ for $i \neq d - 1$ but $R^i f_* O_{X'} \neq 0$ for $i = d-1$. Thus $R^{d-1} f_* O_{X'}$ has codimension $d$ which is not the same as codimension $d-1+2 = d+1$. Thus I think that even for very mild singularities, you can't hope to have the vanishing you want.
In particular, this should mess up the general version of the statement you want, since you can probably localize at the generic point of the non-rational locus.