If not, are there any interesting subcategories that can be concertized? If I am not mistaken, the category of reduced finite type varieties over the complex numbers would be an example, where the forgetful functor to sets would be given by looking at the underlying map of points.



Can the Category of Schemes be Concretized?
The category of schemes is not small-concrete.


Let $S$ be a generating set. Let $U$ be the set of all rings $A \neq 0$ such that $\mathrm{Spec}(A)$ is an open subscheme of a scheme in $S$. Let $X$ be a set whose cardinality is larger than any element of $U$, for example, $2^{\bigsqcup\_{A \in U} A}$. Let $K$ be the field $\mathbb{Q}(t\_x)\_{x \in X}$, where $t\_x$ are a collection of algebraically independent generators indexed by $X$. So $|K|$ is larger than $|A|$ for any $A \in U$. Since ring maps from a field to a nontrivial ring are always injective, $\mathrm{Hom}(\mathrm{Spec}(A),\mathrm{Spec}(K))=\emptyset$ for every $A \in U$, and therefore $\mathrm{Hom}(s,\mathrm{Spec}(K))=\emptyset$ for every $s \in S$.


There is only one map from the empty set to itself. But $\mathrm{Spec}(K)$ has nontrivial isomorphisms, coming from permuting the generators. So 


$\mathrm{Hom}(\mathrm{Spec}(K),\mathrm{Spec}(K)) \longrightarrow \mathrm{Hom}\_{\mathrm{Set}^{S^\mathrm{op}}}( (\mathrm{Spec}(K))(-), (\mathrm{Spec}(K))(-))$


is not injective.


