A classical example : the forgetful functor from topological spaces to sets.
The left adjoint is the "discrete space" functor (sending a set $X$ to the discrete space with underlying space $X$), and the composition just gives the identity on Sets, so clearly Top is not the Eilenberg-Moore category of the monad.
You can see that the forgetful functor does not reflect isomorphisms (a homeomorphism is more than a bijective continuous function), so the monadicity theorem indeed cannot be applied.