I have derived some formula for the lie derivative of a covariant tensor which seems wrong....
Let $X$ is vector field inducing a flow $\phi_t$ and $T$ a rank $2$ covariant tensor, I claim that
$$(\mathcal{L}_XT)(Y,Z)=T(\mathcal{L}_XY,Z)+T(Y,\mathcal{L}_XZ) $$
By definition we have that $$(\mathcal{L}_XT)(Y,Z)=\frac{d}{dt}(\phi_{-t})_*T_{\phi_t(p)}(Y,Z) $$ Now by bi-linearity of $T$ we have:
\begin{align*} (\phi_{-t})_*T_{\phi_t(p)}(Y,Z) &= T((\phi_{-t})*Y_{\phi_t(p)},(\phi_{-t})*Z_{\phi_t(p)}) \\ &=T((\phi_{-t})*Y_{\phi_t(p)}-Y,(\phi_{-t})*Z_{\phi_t(p)})+T(Y,(\phi_{-t})*Z_{\phi_t(p)}) \end{align*} Hence $$\frac{(\phi_{-t})_*T_{\phi_t(p)}(Y,Z)-T_p(Y,Z) }{t}= T(\frac{(\phi_{-t})*Y_{\phi_t(p)}-Y}{t},(\phi_{-t})_*Z_{\phi_t(p)}) +T(Y_p,\frac{(\phi_{-t})*Z_{\phi_t(p)}-Z_p}{t}) $$
We conclude by taking the limit as $t$ goes to $0$ using the continuity of $T$.