Computing cohomology of the logarithmic tangent sheaf $T_X(-\log D)$ is a usually easier once you know the cohomology of the sheaves $T_X, \quad T_X(-D)$ and $N_D$, the normal bundle of $D$ in $X$. To get $H^i(X,T_X(-\log D))$ one can use the exact sequences
$$
0 \to T_X(-D) \to T_X(-\log D) \to T_D \to 0
$$or
$$
0 \to T_X(-\log D) \to TX\to N_{D}\to 0.
$$These sequences make sense even when $D$ is singular (see Sernesi's book on deformation theory). These sheaves and their cohomology groups can be handeled in, say, Macaulay2, if you have the explicit equations.

EDIT: Let me try to explain the above in more detail: If one is in the lucky position that some of the cohomology groups of, say $N_D$, vanish (e.g., a line $L$ on a cubic surface has $h^0(L,N_L)=h^1(L,N_L)=0$, you can calculate the cohomology almost immediately. In the general case, you will have to write down the maps and calculate the cohomology using say, Cech cohomology. Macaulay2 have built-in routines for this, see for example Francesco Polizzi's answer to this question.