When $C$ is essentially small, the presheaf category $[C^\mathrm{op},\mathsf{Set}]$ is the free cocompletion of $C$. The presheaf category $[C^\mathrm{op},\mathsf{Set}]$ is also a topos.
When $C$ is truly large, the presheaf category $[C^\mathrm{op},\mathsf{Set}$] is not even locally small. But the full subcategory $\mathcal{P}C \subset [C^\mathrm{op},\mathsf{Set}]$, consisting of the presheaves which are small colimits of representables, is still the free cocompletion of $C$. When $C$ is small, $\mathcal{P}C$ is the whole category $[C^\mathrm{op},\mathsf{Set}]$, so it is a topos. But when $\mathcal{C}$ is large, is $\mathcal{P}C$ a topos?
Day and Lack, following Rosicky, study when $\mathcal{P}C$ is complete, and when it is cartesian closed. Certainly cartesian closedness is necessary to be a topos; I believe that a cocomplete topos is also complete, so that completeness is also necessary. But I suspect that $\mathcal{P}C$ might only have a subobject classifier if $\mathcal{C}$ is essentially small.
If it makes a difference to restrict to sheaves in some Grothendieck topology, that would be interesting to know. I suppose my motivating example is when $C$ is the category of schemes (or just affine schemes). Algebraic geometers are very interested in sheaves on this category in various topologies; I suspect most such interesting sheaves are small, but I don't know if that restriction gives one a topos to work with.
EDIT
As David Carchedi points out in the comments, I should emphasize that I'm interested in when $\mathcal{P}C$ is an elementary topos (a finitely complete, cartesian closed category with a subobject classifier). As Eric Wofsey argues in the comments, if $C$ is essentially large, then $\mathcal{P}C$ is never a Grothendieck topos (it doesn't have a small generating set).
EDIT
The weakest of observations: the Yoneda embedding $C \to \mathcal{P}C$ is still continuous, so preserves monos, and is fully faithful; so it induces injections on subobject lattices. So if $C$ is not well-powered to begin with, then $\mathcal{P}C$ is also not well-powered, so can't be a topos (given that it's locally small: look at the size of $\mathrm{Hom}(a,\Omega)$).
EDIT
It looks like the canonical way to give a general answer to this problem would be to generalize the work of Menni which characterizes when the exact completion of a finitely complete category is a topos. Taking the category of small sheaves is put in the same framework as exact completions in Mike Shulman's paper which Adeel linked to in the comments. So the way forward ought to be clear. I'm not sure how easy criteria along Menni's lines would be to check in particular cases like schemes, though.
Best Answer
The following argument shows that if $C$ is essentially large (and locally small), then the presheaf $\Omega$ of sieves on $C$ is not small. Unfortunately, as Zhen points out, this does not show that $\mathcal{P}C$ does not have a subobject classifier, since its subobject classifier ought to be the presheaf of small sieves (i.e., sieves such that the corresponding presheaf is a small presheaf). In general, not all sieves are small, and in fact it seems rather difficult to construct any small sieves (exercise: construct a locally small category in which every object has a large set of sieves, but not a single nontrivial small sieve). It thus seems unlikely to me that this argument can be turned into a proof that $\mathcal{P}C$ does not have a subobject classifier without some strong additional hypotheses. Nevertheless, I'm keeping it as an answer in case you may find it somehow helpful (and because it is a nifty argument, even thought it doesn't prove what I hoped it would!).
Suppose that $\Omega$ is a small presheaf. Then every object $a\in C$ has only a small set of sieves, and there is a small set $S$ of objects such that for every $a\in C$ and every sieve $\sigma\in\Omega(a)$, there is an object $s\in S$, a map $f:a\to s$, and a sieve $\tau\in\Omega(s)$ such that $\sigma=f^*\tau$. In fact, if any such $\tau$ exists, then there is a canonical choice, namely $\tau=f_*\sigma$.
For any $a\in C$, let $\sigma_a$ be the sieve consisting of all maps $b\to a$ that do not have a right inverse (this is the second-largest sieve on $a$). We can thus find some $s_a\in S$ and $f_a:a\to s_a$ such that $\sigma_a=f_a^*f_{a*}\sigma_a$. Let's figure out explicitly what the condition $\sigma_a=f_a^*f_{a*}\sigma_a$ means. The sieve $f_{a*}\sigma_a$ consist of all maps of the form $f_ag$ where $g:b\to a$ does not have a right inverse. The sieve $f_a^*f_{a*}\sigma_a$ then consists of all maps $h$ such that $f_ah=f_ag$ for some $g$ that does not have a right inverse. The equation $\sigma_a=f_a^*f_{a*}\sigma_a$ holds iff the latter sieve does not contain the identity $1:a\to a$, or equivalently if every $g:a\to a$ such that $f_a=f_ag$ has a right inverse.
Since $S$ is small and $C$ is essentially large, we can find an essentially large set of objects $D\subseteq C$ such that all the objects $a\in D$ have the same associated object $s_a\in S$. That is, there is a single object $s\in S$ such that for each $a\in D$, there is a map $f_a:a\to s$ such that if $g:a\to a$ is such that $f_a=f_ag$, then $g$ has a right inverse. Now consider the sieves on $s$ generated by these maps $f_a$. Since $\Omega(s)$ is a small set, there is an essentially large set of objects $E\subseteq D$ such that the $f_a$ for $a\in E$ all generate the same sieve on $s$. That is, for any $a,b\in E$, there exist maps $g:a\to b$ and $h:b\to a$ such that $f_bg=f_a$ and $f_ah=f_b$. But then $f_ahg=f_a$, so $hg$ has a right inverse, and similarly $gh$ has a right inverse. In particular, $g$ and $h$ both have right inverses, and so $a$ and $b$ are retracts of each other. But for any object $a$, there are only an essentially small set of objects that are retracts of $a$, and this contradicts the essential largeness of $E$.