How are MTCs permuted by the Galois action on the little disk operad? There is a well-studied action of $\operatorname{Gal}(\bar{\mathbb Q}/\mathbb Q)$ on (some version of) the $E_2$ operad; see for example this MO question.
Modular tensor categories are examples of $E_2$-algebra objects in $\mathrm{Cat}$, the 2-category of linear categories.
How does the $\operatorname{Gal}(\bar{\mathbb Q}/\mathbb Q)$-action on $E_2$ act on the set of MTCs? (I assume it does preserve modularity.) 
To be pedantic, I mean the following. Suppose $(\mathcal C,\otimes,\dots)$ is an MTC. Then we have an action of $E_2$ on the underlying category $\mathcal C$, which I could write as $(\otimes,\dots) : E_2 \to \mathrm{End}(\mathcal C)$. Now choose $\gamma \in \operatorname{Gal}(\bar{\mathbb Q}/\mathbb Q)$, and use it to define an automorphism $\gamma : E_2 \to E_2$. Define a new MTC $\mathcal C^\gamma$ by declaring that the underlying category $\mathcal C$ is unchanged, but the braided tensor structure is modified by $\gamma$ to $(\otimes,\dots) \circ \gamma : E_2 \to E_2 \to \mathrm{End}(\mathcal C)$. I assume that in general $\mathcal{C} \not\cong \mathcal{C}^\gamma$ as braided monoidal categories. So what is the braided monoidal structure on $\mathcal{C}^\gamma$?
In case it makes it easier, I care most about the case where $\gamma$ is in the cyclotomic Galois group $\operatorname{Gal}(\mathbb Q^{cyc}/\mathbb Q)$. Actually, most (all?) MTCs over $\mathbb C$ in fact can be defined over $\mathbb Q^{cyc}$. So it wouldn't surprise me if $\mathcal{C}^\gamma$ only depends on the image of $\gamma$ along the map $\operatorname{Gal}(\bar{\mathbb Q}/\mathbb Q) \to \operatorname{Gal}(\mathbb Q^{cyc}/\mathbb Q)$.
 A: The Galois action on the profinite or pro-$\ell$ completion of $E_2$ comes, as Dan says, from an action of the corresponding version of the Grothendieck-Teichmüller group. However, to the best of my knowledge none of those acts on MTC's. Rather, they act on braided tensor categories which are themselves pro-finite or pro-unipotent in an appropriate sense. However, even in that case although the action might be higly non-trivial this still produces a braided tensor category equivalent to the one you started with (at least if your category is actually ribbon, but I thin this is true in general).
For the pro-algebraic version $GT(k)$, $k$ a field a characteristic 0, those can roughly speaking be identified with those braided tensor categories over $k[[\hbar]]$ which are symmetric monoidal modulo $\hbar$. This includes, e.g., finite dimensional modules over the formal version $U_{\hbar}(\mathfrak g)$ of quantum groups, but typically not their rational or specialized versions.
The Galois action you get for $k=\mathbb{Q}_\ell$ is discussed in some extent at the end of Kassel-Rosso-Turaev's book "Quantum groups and knot invariants", and in Furusho's paper  Galois action on knots II: Proalgebraic string links and knots.
