There are actually some hyperbolic volume results due to Francois Costantino which you can find on his web page

http://www-irma.u-strasbg.fr/~costanti/Papers%20and%20preprints.html

Especially the papers in the Proceedings of the London Math Society and Geometry and Topology.

Also Stavros has an interesting conjecture about the Reshitkhin-Turaev invariant of closed manifolds and the complex Chern-Simons invariant, which includes volume.

http://arxiv.org/abs/0711.1716

Stavros idea is to form a generating function whose coefficients are the Reshetikhin-Turaev invariants of the manifold of level $r$. He proves that the power series converges in a neighborhood of zero in the plane, and then conjectures that the Borel regulator of the manifold has something to do with the poles of the analytic continuation.

More speculatively, using the standard values of $q$, that is $e^{2\pi i/r}$ where $r\geq 3$ is an integer, the Reshetikhin-Turaev invariant of a three-manifold grows polynomially, where the exponent is half the complex dimension of its $SL(2,C)$-character variety, so you don't get exponential growth. This means the asymptotics of the values of the invariants are quite subtle.

On the other hand, by picking other primitive $r$-th roots of unity, the positivity of the quantum dimensions of the representations break down and you can get exponential growth. The failure of positivity is broached in Habegger, Masbaum, Vogel and Blanchet's big paper in Topology on TQFT. Anyways...

I don't know of the results of any experiments about the growth of the Reshetikhin-Turaev invariant with such choices. My guess is that there is interesting stuff going on with the exponential growth rate, that is reflecting the geometry of the underlying manifold.

In a slightly different direction, using "bad" values of $q$ as above, Gregor Masbaum, Jorgen Anderson and Kenji Ueno were able to recover the translation length of an element of the mapping class group of a planar surface with four boundary components on Teichm\"{u}ller space from the representation on the state space assigned to the surface by the TQFT with corners underlying the Reshetikhin-Turaev invariant. They get it as the exponential growth rate of the trace of the induced morphism.

See for instance:

Andersen, Jørgen Ellegaard; Masbaum, Gregor; Ueno, Kenji Topological quantum field theory and the Nielsen-Thurston classification of $M(0,4)$. Math. Proc. Cambridge Philos. Soc. 141 (2006), no. 3, 477--488

Not only has the volume conjecture not been resolved, it is just the tip of the iceberg when it comes to detecting classical geometry from semiclassical limits.