I think you are misunderstanding the role the photon plays as a force mediator in the EM interaction.
Directly to your question: heat is a form of kinetic energy. EM energy is potential energy associated witha field. Force mediators are also associated with fields - is there a heat "field"?
There are four fundamental forces, three have verified associated particles.
EM you know, has the photon, the strong nuclear force has the gluon, and the weak nuclear force has the W and Z bozons.
The remaining force is gravity - the proposed graviton particle has yet to be discovered.
That is reversing the logic.
Photons have energy. All other particles have energy as well. Energy is a property, not an object - it is always the energy of something. There is the energy of the electromagnetic field (which, in some cases, can be expressed as photons), the energy of the electrons, the energy of the fields of the weak interaction, the energy of the fields of the strong interaction, ...
In modern physics, the fields are the fundamental entities. It is possible to express many effects with the presence of particles, but it is not necessary.
The answer depends on what exactly you mean by "heat."
If you mean, "the sensation that you feel when you stand in the sunlight or in front of a fireplace," that is due to energy transferred by electromagnetic radiation from the source to you.
If you mean, "elevated temperature of an object," that is associated with the averaged random component of the kinetic energy of the molecules or atoms that comprise the object.
If you mean, "the sensation that you feel when you touch a hot object", that is due to the transfer of some of the random kinetic energy of molecules in the object to molecules in your hand/finger/whatever.
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