I need to calculate $Tor^*_{\mathcal O_{\mathbb P^3}}(\mathcal O_{L_1}, \mathcal O_{L_2})$, where $L_1$ and $L_2$ are lines on $\mathbb P^3$.
If they are intersecting at a point, I believe that they can be calculated on $\mathbb P^2$ generated by $L_1$ and $L_2$ using the resolution $0 \to \mathcal O_{\mathbb P^2}(-1) \to \mathcal O_{\mathbb P^2} \to \mathcal O_L \to 0$ for both $L=L_i$. But why does $Tor^* (i_* \mathcal F, i_* \mathcal G)=Tor^* (\mathcal F, \mathcal G)$ hold for an inclusion $i$?
Does the fact that the higher $Tor$ functors vanish reflects the fact that $L_1$ and $L_2$ are transversal? This idea is given by the fact that the intersection number can be defined as $$\sum_{p \in L_1 \cap L_2} \sum_{i \geq 0} (-1)^i Tor^i_{\mathcal O_p} (\mathcal F_p, \mathcal G_p),$$ where the higher $Tor$ functors are some 'correction terms'.
If they are not intersecting, should not $Tor$ functors vanish? How are they connected with the local $Tor$ functors?
Is not $\mathcal O_L$ flat over $\mathcal O_{\mathbb P^3}$? Because tensoring by $\mathcal O_l$ is the same as restricting on $L$. So all the higher $\mathcal{Tor}$ sheaf functors vanish, are not they?
Why did you feel in 4., that $\mathcal{O}_L$ is flat over $\mathcal{O}_{\mathbb{P}^3}$? This is false.
For, 3., yes the tors vanish since tor localizes for coherent sheaves.
For the general case, the best path is to use the standard resolution for $L_1$ say. You have, $0\to \mathcal{O}_{\mathbb{P}^3}(-2)\to \mathcal{O}_{\mathbb{P}^3}(-1)^2\to \mathcal{O}_{\mathbb{P}^3}\to\mathcal{O}_{L_1}\to 0$, tensor with $\mathcal{O}_{L_2}$ and since we know all the maps in the above exact sequence, calculate the tors. There are 3 cases, $L_1=L_2$, they intersect at one point, they are disjoint.