Will there there be any interaction between an electric field and a neutral body?
Mathematically, 2 opposite forces which cancel each other is the same thing as no force at all, so there is no physical difference between the two but I understand your question, and here is my answer:
For a particle which isn't made of charged particles, the answer is simple: q=0 so F=0. So neutrino's, for example, don't interact with electric and magnetic fields at all and feel no force at all.
But, an atom which isn't an ion has a total charge of zero but is made of electrons and protons which are charged. This property makes it feel electric fields from neighboring atoms and be repelled / attracted, but this interaction is very week because of the zero net charge.