$Ho(C^T)$ is almost never monadic over $Ho(C)$.  The objects of $Ho(C^T)$ are $T$-algebras in $C$, which means in particular that their $T$-algebra structure commutes strictly, whereas the algebras for the induced monad on $Ho(C)$ will only have algebra structure commuting up to (non-specified, non-coherent) isomorphism.

For instance, if $T$ is the monad derived from an $E_\infty$-operad, then $T$-algebras are $E_\infty$-spaces, whereas $Ho(T)$-algebras are "$H_\infty$-spaces".  These have an obstruction theory specifying when they can be $E_\infty$-ized.