In unpublished notes by Yi Hu (which appear to be no longer online), I found the following:
Corollary 2.4.5. Let the characteristic of $k$ is zero. Assume that a reductive group $G$ acts rationally on a finitely generated $k$-algebra $R$. Let $J$ be an ideal in $R$, invariant under $G$. Then $(R/J)^G = R^G /(J \cap R^G )$.
Essentially, for affine varieties, we can exchange the operations "taking a subvariety" and "taking a quotient".
I have been unable to find a published source for this fact even though it seems fairly basic. I looked in Mumford & Fogarty's book but could not find anything so simple as an affine variety.