The mechanisms behind the exceptional photovoltaic properties of the metallorganic perovskites are still debated and include a ferroelectric (FE) state from the ordering of the electric dipoles of the organic molecules. We present the first anelastic (complex Young's modulus) and new dielectric measurements on CH3NH3PbI3, which provide new insight into the reorientation dynamics of the organic molecules and the reason why they do not form a FE state. The permittivity is fitted within the tetragonal phase with an expression that includes the coupling between FE and octahedral tilt modes, indicating that the coupling is competitive and prevents FE ordering. The onset of the orthorhombic phase is accompanied by sharp stiffening, analogous to the drop of permittivity, due to the hindered molecular dynamics. On further cooling, an intense anelastic relaxation process without a dielectric counterpart suggests the reorientation of clusters of molecules with strong antiferroelectric correlations.