Let $Z \hookrightarrow X$ be a closed subvariety of a smooth projective variety. How do we compute $\mathcal{Tor}_i^{O_X}(O_{Z},O_{Z})$ $(i>0)$ as coherent sheaves on $Z$ where $Z$ is not of complete intersection inside $X$? Are there some good examples?

The completely intersection case is somehow easier using Koszul complex. For instance, $\mathcal{Tor}_1^{O_{\mathbb P^2}}(O_{\mathbb P^1},O_{\mathbb P^1}) \cong O_{\mathbb P^1}(-1)$.