Computing cohomology of the logarithmic tangent sheaf $T_X(-\log D)$ is a usually easier once you know the cohomology of the sheaves
$T_X, \quad T_X(-D)=T_X\otimes I_D$ and $N_D$, the normal bundle of $D$ in $X$. These cohomology groups can be calculated in, say, Macaulay2, if you have the explicit equations. 

To get $H^i(X,T_X(-\log D))$ one can then take cohomology of the exact sequences
$$
0 \to T_X(-D) \to T_X(-\log D) \to T_D \to 0
$$and
$$
0 \to T_X(-\log D) \to TX\to N_{D}\to 0.
$$