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!
The Langlands group is not meant to be the motivic Galois group; rather, it is larger (in Langlands's original formulation), or alternatively not an algebraic group, but a locally compact group which has some kind of underlying algebraic avatar (this is the more recent, indeed current, formulation, due to Kottwitz), so that one can speak of both continuous and algebraic representations.
A toy model to think about is the group $\mathbb C^{\times}$, and the difference between representations of $\mathbb C^{\times}$ just as a topological group, as opposed to $\mathbb C^{\times}$ thought of as a real algebraic group (i.e. thought of as the restriction of scalars of $\mathbb G_m$ from $\mathbb C$ to $\mathbb R$).
To see this example arising in real life, one can think about the difference between arbitrary and algebraic (type $A_0$ in Weil's terminology) Hecke characters for some number field $F$. (This is the theory for $1$-dimensional reps. of the Langlands group $\mathcal L_F$.) The relevant topological group is the idele class group of $L$, while the corresponding algebraic group is what Langlands calls the Serre group in his Ein Maerchen paper (and which is studied, but with different notation and terminology, in Serre's Abelian $\ell$-adic reps. book).
A harder example can be obtained by comparing the global Weil group over a number field to the Taniyama group over this field. This is discussed in Ein Maerchen, and in the book of Deligne, Milne, Ogus, and Shih, Hodge cycles, motives, and Shimura varieties.
(This is the theory obtained by combining $1$-dimensional reps. with finite image reps. of arbitrary dimension.) (See this answer for more on Weil groups.)
Best Answer
Actually, the category of motives isn't equivalent to the category of representations of an affine group scheme except in characteristic zero, and even there the equivalence depends on the choice of a fibre functor.
The functor to motives is supposed to be a universal cohomology theory. Certainly, one would like the target of a cohomology theory to be at least tannakian.
If you assume the Hodge conjecture, then the affine group scheme attached to the category of abelian motives over $\mathbb{C}$ (that generated by abelian varieties) is more-or-less known --- at least its algebraic quotients are classified.
If you assume the Tate conjecture, then the affine groupoid attached to the category of motives over an algebraic closure of $\mathbb{F}_p$ is more-or-less known.
In the general case nothing is known except that the group is VERY BIG --- for example, over $\mathbb{C}$ it has uncountably many distinct quotients isomorphic to PGL(2).