Does the potential energy increase when temperature is raised?
Yes, potential energy increases with increasing temperature for at least the following three reasons:


At a higher temperature, more atoms/molecules are in excited electronic states.  Higher electronic states correspond to greater potential energy. Potential Energy is -2 times Kinetic Energy. So actually, at higher temperature, when more atoms are in higher electronic states, there is more potential energy and less kinetic energy (just considering electronic energy).
At higher temperature, more molecules are in excited vibrational states.  Higher vibrational states correspond to greater potential energy. Half the energy of each vibrational state is kinetic, half potential.
At higher temperature, more molecules are in excited rotational states.  While for a rigid rotor, this would not involve increased potential energy, real molecules are not rigid.  The rotation of a diatomic molecule about an axis perpendicular to the bond stretches the bond as a centrifugal distortion, which represents an increase in potential energy.  


Additionally, for intermolecular forces, to the extent that they involve potentials of the form r^x, the Virial Theorem requires that total energy is divided into potential and kinetic energy in a specific proportion, so total energy can not be increased without increasing potential energy.  

Free translational motion is not subject to a potential, nor is rigid rotation.