I would dare to say that yes, R&S has proved influential in the mathematical sense. At least it made Grothendieck's "Esquisse d'un programme" more visible and it is clear that the topics there like anabelian geometry or the new foundations for homotopical algebra have been two avenues of research of great interest recently. As for "tame topology" my impression is that he topic has not taken off, but I may be wrong about this.

Also it is clear that motives have won a renewed interest since the 90's and the importance of his visions about this (though perhaps not specific details) is amply explained in R&S.

On another front he has expressed the interest in D-modules as a central topic in the cohomology of algebraic varieties together with the philosophy of "six operations" and "cohomological coefficients" that has produced a lot of results and extensions, including, for instance, $p$-adic and logarithmic versions.

A topic that perhaps has not been so intensely pursued is his point of view on the cohomology of singular spaces. According to R&S, there should be a theory o crystals and a theory of co-crystals over any (reasonable) scheme. With smoothness assumptions (over a regular base, say) they should agree (a sort of "Poincaré duality") but on the general case there should be a relationship (related to the nature of singularity). This ideas are presented in a series of footnotes in the 4th part of R&S.

It seems to me that this line of research has not been pursued, mainly for two reasons. Grothendieck himself expressed the possibility of using resolution of singularity and simplicial techniques (or variants) to study the cohomology of a singular variety reducing it to its resolution and resolution of certain open subsets. This was accomplished successfully by Deligne in his "Theorie de Hodge". However the lack of advance in the characteristic $p$ case gives sense to R&S approach, but it seems that mathematicians have other priorities. On the other hand, the big panoply of new objects (algebraic spaces, stacks, derived algebro-geometric objects) possibly has drained people from working on this questions.

Another topic from R&S that has not been addressed is: What is the correct definition of D-module (or crystal) over $\mathrm{Spec}(\mathbb{Z})$? I have no doubt that this is a really hard question to tackle. The advances so far have been small and using a great deal of machinery, I am thinking on the various generalizations of De Rham-Witt theory to mixed characteristic situations.

I would dare to say that yes, R&S has proved influential in the mathematical sense. At least it made Grothendieck's "Esquisse d'un programme" more visible and it is clear that the topics there like anabelian geometry or the new foundations for homotopical algebra have been two avenues of research of great interest recently. As for "tame topology" my impression is that he topic has not taken off, but I may be wrong about this.[Edit: I am wrong: see Thierry Zell comment after this.]

Also it is clear that motives have won a renewed interest since the 90's and the importance of his visions about this (though perhaps not specific details) is amply explained in R&S.

On another front he has expressed the interest in D-modules as a central topic in the cohomology of algebraic varieties together with the philosophy of "six operations" and "cohomological coefficients" that has produced a lot of results and extensions, including, for instance, $p$-adic and logarithmic versions.

A topic that perhaps has not been so intensely pursued is his point of view on the cohomology of singular spaces. According to R&S, there should be a theory o crystals and a theory of co-crystals over any (reasonable) scheme. With smoothness assumptions (over a regular base, say) they should agree (a sort of "Poincaré duality") but on the general case there should be a relationship (related to the nature of singularity). This ideas are presented in a series of footnotes in the 4th part of R&S.

It seems to me that this line of research has not been pursued, mainly for two reasons. Grothendieck himself expressed the possibility of using resolution of singularity and simplicial techniques (or variants) to study the cohomology of a singular variety reducing it to its resolution and resolution of certain open subsets. This was accomplished successfully by Deligne in his "Theorie de Hodge". However the lack of advance in the characteristic $p$ case gives sense to R&S approach, but it seems that mathematicians have other priorities. On the other hand, the big panoply of new objects (algebraic spaces, stacks, derived algebro-geometric objects) possibly has drained people from working on this questions.

Another topic from R&S that has not been addressed is: What is the correct definition of D-module (or crystal) over $\mathrm{Spec}(\mathbb{Z})$? I have no doubt that this is a really hard question to tackle. The advances so far have been small and using a great deal of machinery, I am thinking on the various generalizations of De Rham-Witt theory to mixed characteristic situations.