If an inverse covariance matrix is sparse, what can I say about the covariance matrix? How does the sparsity condition on an inverse covariance matrix affect the actual covariance matrix?
 A: As already commented by Yair there is no specific sparsity condition of the inverse covariance matrix that affects the actual covariance matrix or vice versa. Anything other than trivial sparsity matrix patterns (ie. diagonal) have no guarantee that they will reflected on both a particular matrix and its inverse. Even tridiagonal matrices can easily have non sparse inverses.
For particular cases where the sparsity of the matrix occurs in blocks you might be able to derive some results stemming from the Block matrix pseudoinverse algorithm which states that: 
$\left[\begin{array}{cc} A & B \\C & D \end{array}\right]^{-1} =\begin{bmatrix}
                 (A - BD^{-1}C)^{-1}         & -A^{-1}B(D - CA^{-1}B)^{-1} \\
                 -D^{-1}C(A - BD^{-1}C)^{-1} & (D - CA^{-1}B)^{-1}
\end{bmatrix}$
but that's probably about it (purely anecdotally, I have tried to impose sparsity patterns through the Cholesky decomposition of a PSD matrix but I failed in my trial-and-error foray). You might also want to consider looking into the Cuthill–McKee algorithm (CM) if you expect some adjacency feature to be reflected in the covariance matrix. The CM algorithm permutes a sparse matrix that has a symmetric sparsity pattern into a band matrix form with a small bandwidth, this might help preserve some sparsity towards the off-diagonal entries of the inverse matrix but that is not guaranteed. (Applying CM -if reasonable- can be very helpful for particular applications (eg. in 2D smoothing routines) and may significantly speed-up your computations.)
