Well speed of light is a constant therefore in order to reduce the value of E to 0 we have to reduce its rest mass to 0, as nobody has its mass as 0 it is not possible for E to be 0, you might argue that the rest mass of photon is zero, well it might be true but, the equation is valid only for particles at rest, in general the mass energy equation is given as:
E2=m20C4+p2c2
When the rest mass is 0 the energy comes from its linear momentum, therefore the value of E is never equal to zero.