This seems like a natural question to ask, but I've not seen it discussed in my reading around (limited to Easton's paper, the third edition of Jech's Set theory and a small handful of articles). What do you get when forcing with a proper class of conditions that you don't get when forcing with, say, a set of conditions larger than one's model of set theory? Or rather, why isn't a large set of conditions enough?
Blass makes a throwaway remark in his 1984 paper The interaction between category theory and set theory
Although this approach [reflection principles] was first proposed in connection with the problem of foundations for category theory, it is natural to use it whenever objects seem to be too large to be coded as sets. In particular, it seems to me that it should be of some use in clarifying forcing with proper classes by making the natural (regular open) Boolean algebra available even though it is superlarge.
It seems to indicate that if we accept some sort of reflection principle, use an innaccessible cardinal $\kappa$ (or similar) – hence a Grothendieck universe – and a set of forcing conditions larger than $\kappa$, then we should arrive at our goal without using a proper class of conditions.
Alternatively, cannot one (ok, this is very naive, but this is why I'm asking) consider an inaccessible in ZFC and thus cook up a model of NBG, and then work with that a la Easton – and then at the end turn around a say 'Ahah! I was working in ZFC the whole time!'
One reason I ask is that in the paper Injectivity, projectivity, and the axiom of choice, Blass gives a symmetric model of ZFA with no nontrivial injective abelian groups using an uncountable set of atoms and a base model of ZFCA whose sets were in some sense 'small' (they arise, if I understand correctly, using the cumulative hierarchy generated from $A$ in the usual sense, but only taking countable subsets of $A$ at the first stage, rather than all of $\mathscr{P}A$). However, he gives a model of ZF with no nontrivial injective abelian groups using forcing involving a proper class of conditions. (Notice that Jech-Sochor is not useful in its usual statement because a global statement about a proper class of objects is required.) Perhaps the techniques given in Blass' Theorem 3.2 have been given a general treatment by now, I do not know.
Best Answer
The idea behind the remark quoted in the question was that, in situations ordinarily treated with proper-class forcing (e.g., Easton's theorem), the work can be transcribed rather routinely into a Feferman-style set theory (ZFC plus a constant $\kappa$ for an ordinal and axioms saying, one formula at a time, that $V_\kappa$ is an elementary submodel of the universe $V$). Just do with $V_\kappa$ what you would otherwise have done with $V$. Where (for example) Easton got arbitrary cardinal exponentiation at all regular cardinals, you'd now get arbitrary cardinal exponentiation only at all regular cardinals below $\kappa$, but that's "morally" or "intuitively" the same (and gives the same relative consistency result) because of the elementarity of $V_\kappa$ in $V$. This "large set" approach allows you to work with the framework of Boolean-valued models rather than forcing, whereas a proper-class forcing would, in general, need super-classes (yet another level higher in the cumulative hierarchy) to do this.
In both frameworks, the real issue is not whether you work with large (i.e., $\kappa$-sized or bigger) sets or with proper classes but rather what additional conditions you impose on your forcing notions (or Boolean-valued models). As Nate pointed out, you need some conditions (in either framework) to make sure you get a model of ZFC. If you just go blindly ahead (in either forcing), you could, for example, add a proper class (respectively a $\kappa$-sized family) of Cohen reals, so that the continuum will no longer be a set in your forcing extension (of $V$, respectively $V_\kappa$). Or you might collapse all the cardinals (resp. all the cardinals below $\kappa$).
Of course, some people might want to sacrifice (part of) ZFC and work with such "strange" models. If I remember correctly, Rudy Rucker once (before he turned to science-fiction writing) proposed working in the theory obtained from ZFC by deleting the power set axiom and adding Martin's Axiom for arbitrarily large collections of dense sets (so the continuum has to be a proper class). But here again, it seems to me that it makes little difference which framework you use.
Also, I recall that Sy Friedman did some work on super-class forcing. I don't know any of the details, but I would expect that this too can be easily recast in terms of forcing over a Feferman-style $V_\kappa$.
Finally, let me mention that, if you force in the Feferman framework, you actually have two choices for what should be the generic extension of $V_\kappa$. One is to take the elements of rank below $\kappa$ in the generic extension of the full universe. The other is to take the denotations of names whose rank is below $\kappa$. The two seem to coincide in nice cases, but I don't see any reason for them to coincide in general. (The second is what corresponds to proper-class forcing over $V$.)