The basic contention here is that Wiles' work uses cohomology of sheaves on certain Grothendieck topologies, the general theory of which was first developed in Grothendieck's SGAIV and which requires the existence of an uncountable Grothendieck universe. It has since been clarified that the existence of such a thing is equivalent to the existence of an inaccessible cardinal, and that this existence -- or even the consistency of the existence of an inaccessible cardinal! -- cannot be proved from ZFC alone, assuming that ZFC is consistent.
Many working arithmetic and algebraic geometers however take it as an article of faith that in any use of Grothendieck cohomology theories to solve a "reasonable problem", the appeal to the universe axiom can be bypassed. Doubtless this faith is predicated on abetted by the fact that most arithmetic and algebraic geometers (present company included) are not really conversant or willing to wade into the intricacies of set theory. I have not really thought about such things myself so have no independent opinion, but I have heard from one or two mathematicians that I respect that removing this set-theoretic edifice is not as straightforward as one might think. (Added: here I meant removing it from general constructions, not just from applications to some particular number-theoretic result. And I wasn't thinking solely about the small etale site -- see e.g. the comments on crystalline stuff below.)
Here is an article which gives more details on the matter:
Note that I do not necessarily endorse the claims in this article, although I think there is something to the idea that written work by number theorists and algebraic geometers usually does not discuss what set-theoretic assumptions are necessary for the results to hold, so that when a generic mathematician tries to trace this back through standard references, there may seem to be at least an apparent dependency on Grothendieck universes.
P.S.: If a mathematician of the caliber of Y.I. Manin made a point of asking in public whether the proof of the Weil conjectures depends in some essential way on inaccessible cardinals, is this not a sign that "Of course not; don't be stupid" may not be the most helpful reply?
Though there are several automorphic papers discussing the Tannakian outlook (notably Ramakrishnan's article in Motives (Seattle 1991, AMS) and Arthur's A note on the Langlands group, (referred to above) there is as yet no formulation of Langlands correspondence between Galois representations and automorphic representations as an equivalence of Tannakian categories. There are (at least) two outstanding fundamental questions on the Tannakian aspects of the Langlands correspondence.
1) What is the definition of the category of automorphic representations for any number field?
here one means automorphic representations for GL, any $n \ge 0$.
2) How to endow the category in 1) with a tensor structure so as to render it Tannakian?
here the postulated Tannakian group is the "Langlands Group" which is much larger
than the motivic Galois group (not all automorphic representations correspond to Galois
representations..only algebraic ones do - see work of Clozel and more recent work of Buzzard-Gee).
An interesting point: Arthur's paper postulates the Langlands group as an extension of the usual
Galois group by a (pro-) locally compact group whereas the motivic Galois group is an extension
of the usual Galois group by a pro-algebraic group. An illustration of the difference is provided
by the case of abelian motives; the Langlands group is the abelianisation of the Weil group
whereas the motivic group is the Taniyama group (see references below).
But the Tannakian outlook, despite its present inaccessibility, has already made a profound impact. See Langlands paper "Ein Marchen etc" (where the Tannakian aspect was first written out with many consequences for the Taniyama group (Milne's notes)) as well as
Serre's book Abelian l-adic representations for many references.
Nothing seems to be known regarding the second question. However, see (page 6 of) these comments of Langlands:
"Although there is little point in premature speculation about the form that the final theory connecting automorphic forms and motives will take, some anticipation of the possibilities has turned out to be useful. Motivic $L$-functions, in terms of which Hasse-Weil zeta functions are expressed, are introduced in a Tannakian context.
....
An adequate Tannakian formulation of functoriality and of the relation between automorphic representations and motives ([Cl1, Ram]) will presumably include the Tate conjecture ( [Ta] ) as an assertion of surjectivity. The Tate conjecture itself is intimately related to the Hodge conjecture whose formulation is algebro-geometrical and topological rather than arthmetical. ..."
The references here are to Clozel and Ramakrishnan's papers and then Tate's paper for the Tate conjecture.
This is just a rough answer from a novice..for a precise and detailed answer, let us wait for the experts!
Best Answer
"Langlands for $n = 2$", to the extent that such a notion is defined, is more than just Shimura--Taniyama, and for even Galois representations/Maass forms, it is still very much open. (See here for more on this.) For odd Galois representations of dimension $2$, though, it is completely (or almost completely, depending on exactly what you mean by "Langlands") resolved at this point, with the proof of Serre's conjecture (by Khare, Wintenberger, and Kisin) playing a pivotal role.
Much is known for $n > 2$ (see the web-pages of e.g. Michael Harris, Richard Taylor, and Toby Gee). A key point is that it is hard to say anything outside the essentially self-dual case (and this is a condition which is automatic for $n = 2$). A second is that Serre's conjecture is not known in general.
If one restricts to the regular (corresponding to weight $k \geq 2$ when $n = 2$), essentially self-dual case (automatic when $n = 2$), then basically everything for $n = 2$ carries over to $n > 2$, with the exception of Serre's conjecture. (See e.g. the recent preprint of Barnet-Lamb--Gee--Geraghty--Taylor.)
So really, what is special for $n = 2$ is that Serre's conjecture was able to be resolved. And the reason that this has (so far) been possible only for $n = 2$ is that the proof depends on certain special facts about $2$-dimensional Galois representations.
More specifially:
In the particular case of Shimura--Taniyama, the Langlands--Tunnell theorem allowed Wiles to resolve a particular case of Serre's conjecture (for $p = 3$). To then get all the necessary cases of Serre's conjecture, Wiles introduced the $3$-$5$ switch.
The general proof of Serre's conjecture uses a massive generalization of the $3$-$5$ switch (along with many other techniques), and although (unlike with Wiles's argument) it doesn't build specifically on Langlands--Tunnell, it does build on a result of Tate which is a special fact about $2$-dimensional representations of $G_{\mathbb Q}$ over a finite field of characteristic $2$.