It would be useful, if you'd point out the precise "open questions" regarding these evaluations that are of interest to you.

But since you mentioned Plamen Koev's work, I am sure you have tried out his matlab code [for computing Hypergeometric functions of matrix argument][1]. For the particular case of the normalization constant of the Bingham distribution, I am sure methods for approximating high-dimensional integrals numerically will prove to be effective, because, although scary looking, the normalization constant still has a very nice form:

\begin{equation*}
  {}_1F_1(\frac12,\frac p2, A) := \int_{S^{p-1}} e^{x^TAx}dx,
\end{equation*}
where the integration is wrt to the uniform distribution on the unit hypersphere $S^{p-1}$.

(PS: This integral has previously been discussed on MO, e.g., [here in this question of L. Nicolaescu][2])

**Edit:** Regarding numerical approximation. Here is what seems to be the latest in this direction. Have a look at the [geodesic monte carlo sampling][3] method of Byrne and Girolami; (they discuss sampling from the Bingham distribution). Once you have that, then it should be "easy" to estimate the normalization constant. But I guess the bad thing might again be a lack of guarantees on how long it takes to get a given accuracy approximation---but for now, seems like the abovecited approach might be the most promising.


  [1]: http://math.mit.edu/~plamen/software/mhgref.html
  [2]: https://mathoverflow.net/questions/104919/can-one-recognize-this-symmetric-function/104935#104935
  [3]: http://arxiv.org/abs/1301.6064