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 $$ 0 \to T_X(-\log D) \to TX\to N_{D}\to 0. $$