If you'll allow me to abstract away from $Top$ for a minute, there are many examples of quotient-reflective subcategories. A simple example is that abelian groups are quotient-reflective in groups, because the unit of the adjunction is a quotient map $\pi: G \to G/[G, G]$. In general, if you start with an algebraic theory $T$ whose signature is given by a set of function symbols, and if $T'$ has the same signature but more universally quantified equational axioms in addition to those already in $T$, then the category of $T'$-models, as a full subcategory of the category of $T$-models, is quotient-reflective: if $G$ is a $T$-model, then the unit of the adjunction is the quotient of $G$ by the smallest $T$-closed congruence which contains pairs of terms whose equality is asserted in $T'$.

A slightly different example is that sheaves (w.r.t. any site) are quotient-reflective in the category of presheaves on the underlying category of the site. (Actually, this may be somewhat similar, as one sheafifies by applying the plus construction twice, and applying the plus construction once reflects into separated presheaves, which are defined by extra equations on presheaves.)

The examples you give in $Top$ are interesting: in each of the $T_0$ and $T_2$ cases, the extra condition that defines the full subcategory asserts an equational conclusion. Thus, $T_0$ says $x = y$ if $x$ and $y$ have the same neighborhoods; $T_2$ says that $x = y$ if every open of $(x, y)$ in $X \times X$ contains a point $(z, z)$. They do *not* assert an existential conclusion on points as say compactness would (e.g., every ultrafilter converges to *some* point), which would require adjoining extra points when one passes to the reflection, in order to witness such existence. So it seems a useful criterion for quotient-reflectivity for concrete categories of models is that the full subcategory is defined by extra universally quantified equational conditions in the language used to specify the ambient category, just as in the algebraic examples above.

(By the way, I am intepreting "quotient" to mean "regular epi", i.e., a map that is the coequalizer of its own kernel pair, and I interpret "subcategory" here to mean "full subcategory, as is usual when discussing reflective subcategories. I'm not sure I can give a more abstract nonsense reply right now, but this should give a useful class of examples.)